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OSMANIA UNIVERSITY LIBRARY 

Call No.S 5.VJ / I Accession No. 11 I? 
Author ffli*lLtfcv. \iJ - # 



This book should be returned on or before the date 
last marked below. |) ^^^ 



THE UNIVERSE IN THE LIGHT 
OF MODERN PHYSICS 



THE UNIVERSE IN THE LIGHT 
OF MODERN PHYSICS 

by 

DR. MAX PLAJiCJC 

Professor of Theoretical Physics at the 
University of Berlin 

Translated by 
W. H. JOHNSTON, B.A. 



LONDON 

GEORGE ALLEN & UNWIN LTD 
MUSEUM STREET 



The present work is a translati\ 
der neuen Physik" and "P" 
neuerer Forschung," published by Jcfti j?. Ba\ 
The two works have been runUnto oij, the 
mencing \n p. 58. 




FIRST PUBLISHED 



ENGLISH 1931 



THE UNIVERSE IN THE LIGHT 
OF MODERN PHYSICS 



PHYSICS is an exact Science and hence depends 
upon measurement, while all measurement itself 
requires sense-perception. Consequently all the 
ideas employed in Physics are derived from the 
world of sense-perception. It follows from this 
that the laws of Physics ultimately refer to 
events in the world of the senses; and in view 
of this fact many scientists and philosophers 
tend to the belief that at bottom Physics is con- 
cerned exclusively with this particular world. 
What they have in mind, of course, is the world 
of man's senses. On this view, for example, what 
is called an "Object" in ordinary parlance is, 
when regarded from the standpoint of Physics, 
simply a combination of different sense-data 
localized in one place. It is worth pointing out 
that this view cannot be refuted by logic, since 
logic itself is unable to lead us beyond the con- 
fines of our own senses; it cannot even compel 
one to admit the independent existence of others 
outside oneself, 



In Physics, however, as in every other science, 
common sense alone is not supreme; there must 
also be a place for Reason. Further, the mere 
absence of logical contradiction does not neces- 
sarily imply that everything is reasonable. Now 
reason tells us that if we turn our back upon a 
so-called object and cease to attend to it, the 
object still continues to exist. Reason tells us 
further that both the individual man and man- 
kind as a whole, together with the entire world 
which we apprehend through our senses, is no 
more than a tiny fragment in the vastness of 
Nature, whose laws are in no way affected by 
any human brain. On the contrary, they existed 
long before there was any life on earth, and will 
continue to exist long after the last physicist 
has perished. 

It is considerations of this kind, and not any 
logical argument, that compel us to assume the 
existence of another world of reality behind the 
world of the senses ; a world which has existence 
independent of man, and which can only be per- 
ceived indirectly through the medium of the 
world of the senses, and by means of certain 
symbols which our senses allow us to apprehend. 
It is as though we were compelled to contemplate 
a certain object in which we are interested 



through spectacles of whose optical properties 
we were entirely ignorant. 

If the reader experiences difficulty in following 
this argument, and finds himself unable to accept 
the idea of a real world which at the same time 
is expressly asserted to lie beyond our senses, 
we might point out that there is a vast differ- 
ence between a physical theory complete in every 
detail, and the construction of such a theory. 
In the former case the content of the theory can 
be analysed exactly, so that it is possible to 
prove at every point that the notions which we 
apply to the world of sense are adequate to the 
formulation of this theory; in the latter case we 
must develop a theory from a number of indi- 
vidual measurements. The second problem is 
very much more difficult, while the history of 
Physics shows that whenever it has been solved, 
this has been done on the assumption of a real 
world independent of our senses; and it seems 
reasonably certain that this will continue to be 
the case in the future. 

But besides the world of sense and the real 
world, there is also a third world which must be 
carefully distinguished from these: this is the 
world of Physics. It differs from the two others 
because it is a deliberate hypothesis put forward 



10 

by a finite human mind; and as such, it is sub- 
ject to change and to a kind of evolution. Thus 
the function of this world of Physics may be 
described in two ways, according as it is related 
to the real world, or to the world of the senses. 
In the first case the problem is to apprehend the 
real world as completely as possible; in the second, 
to describe the world of the senses in the simplest 
possible terms. There is no need, however, to 
assign superior merit to either of these formula- 
tions, since each of them, taken by itself alone, 
is incomplete and unsatisfactory. On the one 
hand, the real world cannot be apprehended 
directly at all; while on the other no definite 
answer is possible to the question : Which is the 
simplest description of a given number of inter- 
dependent sense-perceptions? In the history of 
Physics it has happened more than once that, 
of two descriptions, one was for a time considered 
the more complicated but was later discovered 
to be the simpler of the two. 

The essential point therefore is that these two 
formulations of the problem, when practically 
applied, shall be complementary to each other 
and not contradictory. The first is an indispen- 
sable aid to the groping imagination of the 
investigator, supplying him with ideas without 



which his work remains unfruitful; the second 
provides him with a firm foundation of facts. 
In actual practice individual physicists are influ- 
enced in their investigations by their personal 
preference for metaphysical, or for positivist, 
ideas. But besides the metaphysicians and the 
positivists there is a third group of students who 
investigate the world from the physical point 
of view. They differ from the first two groups in 
being interested not so much in the relation 
between the world of physics on the one hand, 
and the real world and the world of sense-data 
on the other, as in the internal consistency and 
logical structure of the world of physics. These 
men form the axiomatic school, whose activity 
is as necessary and useful as is that of the others. 
At the same time, they are equally exposed to 
the danger of specialization which, in their case, 
would lead to a barren formalism taking the 
place of a fuller understanding of the world of 
Physics. For as soon as contact with reality has 
been lost, physical law ceases to be felt as the 
relation between a number of magnitudes which 
have been ascertained independently of one 
another, and becomes a mere definition by which 
one of these magnitudes is derived from the 
others. In this method there is a particular 



12 

attraction, due to the fact that a physical magni- 
tude can be defined far more exactly by means 
of an equation than by means of measurement. 
But at the same time, this method amounts to 
a renunciation of the true meaning of magni- 
tude; while it must also be remembered that 
confusion and misunderstanding result when the 
same name is retained in order to denote a 
changed meaning. 

We see, then, how physicists are at work in 
different directions and from different standpoints 
in elaborating a systematic view of the world 
of Physics. Nevertheless the aim of all these 
endeavours is the same, and consists in estab- 
lishing a law which connects the events of the 
world of sense with one another and with those 
of the real world. Naturally, these different 
tendencies predominated in turn at different 
stages of history. Whenever the physical world pre- 
sented a stable appearance, as in the second half 
of the last century, the metaphysical view tended 
to predominate, and it was believed that a com- 
plete grasp of the real world was relatively near. 
Conversely, in times of change and insecurity 
like the present, positivism tends to occupy the 
foreground; for in such times a careful student 
will tend to seek support where he can find real 



security; and this is to be found precisely in the 
events of the world of the senses. 

Now if we consider the different forms which 
the view of the physical world has taken in the 
course of history, and if we look for the peculiar- 
ities which characterized these changes, two facts 
will strike us with special force. First, it is plain 
that when regarded as a whole, all the changes 
in the different views of the world of Physics 
do not constitute a rhythmical swing of the 
pendulum. On the contrary, we find a clear 
course of evolution making more or less steady 
progress in a definite direction; progress which 
is best described by saying that it adds to the 
content of the world of sense, rendering our know- 
ledge more profound and giving us a firmer grasp 
of it. The most striking instance of this is found 
in the practical application of Physics. Not even 
the most confirmed sceptic can deny that we see 
and hear at a greater distance and command 
greater forces and speeds than an earlier genera- 
tion; while it is equally certain that this progress 
is an enduring increase of knowledge, which is 
in no danger of being described as an error and 
rejected at any future date. 

Secondly, it is a very striking fact that the 
impulse towards simplification and improvement 



14 

of the world-picture of Physics was due in each 
instance to some kind of novel observation 
that is, to some event in the world of sense. 
But at the same moment the structure of this 
physical world consistently moved farther and 
farther away from the world of sense and lost 
its former anthropomorphic character. Still fur- 
ther, physical sensations have been progressively 
eliminated, as for example in physical optics, in 
which the human eye no longer plays any part 
at all. Thus the physical world has become pro- 
gressively more and more abstract ; purely formal 
mathematical operations play a growing part, 
while qualitative differences tend to be explained 
more and more by means of quantitative differ- 
ences. 

Now we have already pointed out that the 
physical view of the world has been continually 
perfected and also related to the world of 
sense. If this fact is added to those mentioned 
in the last paragraph, the result is extraordinarily 
striking; at first, indeed, it appears completely 
paradoxical. Of this apparent paradox there is, 
in my opinion, only one rational explanation. 
This consists in saying that as the view of the 
physical world is perfected, it simultaneously 
recedes from the world of sense; and this process 



15 

is tantamount to an approach to the world of 
reality. I have no logical proof on which to base 
this opinion; it is impossible to demonstrate the 
existence of the real world by purely rational 
methods: but at the same time it is equally 
impossible ever to refute it by logical methods. 
The final decision must rest upon a common- 
sense view of the world, and the old maxim still 
remains true that that world-view is the best 
which is the most fruitful. Physics would occupy 
an exceptional position among all the other 
sciences if it did not recognize the rule that the 
most far-reaching and valuable results of investi- 
gation can only be obtained by following a road 
leading to a goal which is theoretically unobtain- 
able. This goal is the apprehension of true reality. 



What changes have taken place in the physical 
view of the world during the last twenty years? 
We all know that the changes which have occurred 
during this period are among the most profound 
that have ever arisen in the evolution of any 
science; we also know that the process of change 
has not yet come to an end. Nevertheless it 



i6 

would appear that in this flux of change certain 
characteristic forms of the structure of this 
new world are beginning to crystallize; and it 
is certainly worth while to attempt a description 
of these forms, if only in order to suggest certain 
improvements. 

If we compare the old theory with the new, 
we find that the process of tracing back all 
qualitative distinctions to quantitative distinc- 
tions has been advanced very considerably. All 
the various chemical phenomena, for example, 
have now been explained by numerical and 
spatial relations. According to the modern view 
there are no more than two ultimate substances, 
namely positive and negative electricity. Each 
of these consists of a number of minute particles, 
similar in nature and with similar charges of an 
opposite character; the positive particle is called 
the proton, the negative the electron. Every 
chemical atom that is electrically neutral con- 
sists of a number of protons cohering with one 
another, and of a similar number of electrons, 
some of which are firmly fixed to the protons, 
together with which they form the nucleus of 
the atom, while the rest revolve around the 
nucleus. 

Thus the Hydrogen atom, the smallest of all, 



17 

has one proton for nucleus and one electron 
revolving round the nucleus; while the largest 
atom, Uranium, contains 238 protons and 238 
electrons; but only 92 electrons revolve round 
the nucleus while the others are fixed in it. 
Between these two atoms lie all the other ele- 
ments, with many kinds of different combinations. 
The chemical properties of an element depend, 
not on the total number of its protons or electrons, 
but on the number of revolving electrons, which 
yield the atomic number of the element. 

Apart from this important advance, which is 
however merely the successful application of an 
idea first evolved many centuries ago, there are 
two completely new ideas which distinguish the 
modern conception of the world from its pre- 
decessor; these are the Theory of Relativity, and 
the Quantum Theory. It is these two ideas 
which are peculiarly characteristic of the new 
world of Physics. The fact that they appeared 
in science almost simultaneously is something 
of a coincidence; for their content, as well 
as their practical effect upon the structure of 
the physical view of the world, are entirely 
different. 

The Theory of Relativity seemed at first to 
introduce a certain amount of confusion into the 

Modern Physics _ 

D 



i8 

traditional ideas of Time and Space; in the long 
run, however, it has proved to be the completion 
and culmination of the structure of classical 
Physics. To express the positive result of the 
Special Theory of Relativity in a single word, 
it might be described as the fusion of Time and 
Space in one unitary concept. It is not, of course, 
asserted that Time and Space are absolutely 
similar in nature; their relation resembles that 
between a real number and an imaginary number, 
when these are combined together to form the 
unified concept of a complex number. Looked 
at in this way, Einstein's work for Physics 
closely resembles that of Gauss for Mathematics. 
We might further continue the comparison by 
saying that the transition from the Special to 
the General Theory of Relativity is the counter- 
part in Physics to the transition from linear 
functions to the general theory of functions in 
mathematics. 

Few comparisons are entirely exact, and the 
present is no exception to the rule. At the same 
time it gives a good idea of the fact that the 
introduction of the Theory of Relativity into the 
physical view of the world is one of the most 
important steps towards conferring unity and 
completeness. This appears clearly in the results 



ig 

of the Theory of Relativity, especially in the 
fusing of momentum and energy, in the identifi- 
cation of the concept of mass with the concept 
of energy, of inertial with ponderable mass, and 
in the reduction of the laws of gravitation to 
Riemann's geometry. 

Brief though these main outlines are, they 
contain a vast mass of new knowledge. The new 
ideas mentioned apply to all natural events great 
and small, beginning with radio-active atoms 
emanating waves and corpuscles, and ending 
with the movements of celestial bodies millions 
of light-years away. 

The last word on the Theory of Relativity 
probably still remains to be said. Surprises may 
yet await us, especially when we consider that 
the problem of amalgamating Electrodynamics 
with Mechanics has not yet been definitely solved. 
Again, the cosmological implications of the 
Theory of Relativity have not yet been fully 
cleared up, the chief reason being that every- 
thing depends upon the question whether or not 
the matter of outer space possesses a finite 
density; this question has not yet been answered. 
But whatever reply is eventually given to these 
questions, nothing will alter the fact that the 
Principle of Relativity has advanced the classical 



20 

physical theory to its highest stage of completion, 
and that its world-view is rounded off in a very 
satisfactory manner. 

This fact will perhaps be a sufficient reason 
for devoting no more time to the Theory of 
Relativity; I might also point out that there are 
many treatises on the Theory adapted to the 
requirements of readers of every kind. 



3 



The idea of the universe as thus far described 
appeared almost perfectly adapted to its pur- 
pose; but this state of affairs has suddenly been 
upset by the Quantum Theory. Here again I 
shall attempt to describe the characteristic idea 
of this hypothesis in one word. We may say, 
then, that its essence consists in the fact that it 
introduces a new and universal constant, namely 
the elementary Quantum of Action. It was this 
constant which, like a new and mysterious 
messenger from the real world, insisted on turn- 
ing up in every kind of measurement, and con- 
tinued to claim a place for itself. On the other 
hand, it seemed so incompatible with the tradi- 
tional view of the universe provided by Physics 



21 

that it eventually destroyed the framework of 
this older view. 

For a time it seemed that a complete collapse 
of classical Physics was not beyond the bounds 
of possibility; gradually however it appeared, as 
had been confidently expected by all who believed 
in the steady advance of science, that the intro- 
duction of the Quantum Theory led not to the 
destruction of Physics, but to a somewhat pro- 
found reconstruction, in the course of which the 
whole science was rendered more universal. For 
if the Quantum of Action is assumed to be 
infinitely small, Quantum Physics becomes merged 
in classical Physics. In fact the foundations of 
the structure of classical Physics not only proved 
unshakable, but actually were rendered firmer 
through the incorporation of the new ideas. The 
best course, therefore, will be first to examine 
the latter. 

It will be best to begin by enumerating the 
essential component features. These are the 
universal constants, e.g. the gravitational con- 
stant, the velocity of light, the mass and charge 
of electrons and protons. These are perhaps the 
most tangible symbols of a real world, and they 
retain their meaning unchanged in the new view 
of the universe. Further, we may mention the 



22 

great principles of the conservation of energy 
and of momentum, which, although they were 
under suspicion for a time, have eventually 
emerged unimpaired. It should be emphasized 
that in this process of transition these principles 
were proved to be something more than mere 
definitions, as some members of the Axiomatic 
School would like to believe. Further, we may 
mention the main laws of thermodynamics, and 
especially the second law, which through the 
introduction of an absolute value for entropy 
obtained a more exact formulation than it 
possessed in classical Physics. Lastly we may 
point to the Principle of Relativity, which has 
proved itself a reliable and eloquent guide in the 
new regions of Quantum Physics. 

The question may now be asked whether 
modern Physics differs at all from the older 
Physics, if all these foundations of classical 
Physics have remained untouched. It is easy to 
find an answer to this question by examining 
the elementary Quantum of Action somewhat 
more closely. It implies that in principle an 
equation can be established between energy and 
frequency; E = hv. 1 It is this equation which 

1 In this equation E stands for Energy, and v for Fre- 
quency, that is the number of vibrations per second. 



23 

classical Physics utterly fails to explain. The 
fact itself is so baffling because energy and fre- 
quency possess different dimensions; energy is a 
dynamic magnitude, whereas frequency is a kine- 
matic magnitude. This fact in itself, however, 
does not contain a contradiction. The Quantum 
Theory postulates a direct connection between 
dynamics and kinematics; this connection is due 
to the fact that the unit of energy, and conse- 
quently the unit of mass, are based upon the 
units of length and of time; thus the connection, 
so far from being a contradiction, enriches and 
rounds off the classical theory. There is, neverthe- 
less, a direct contradiction, which renders the new 
theory incompatible with the classical theory. 
The following considerations make clear this 
contradiction. Frequency is a local magnitude, 
and has a definite meaning only for a certain 
point in space; this is true alike of mechanical, 
electric and magnetic vibrations, so that all that 
is requisite is to observe the point in question 
for a sufficient time. Energy on the other hand 

For example, light vibrations range from about 400 
million million per second to about 800 million million. 
h represents "Planck's Constant", discovered by the 
author of this work. It is an unchanging or invariable 
quantity, and extremely minute, its value being 655 
preceded by 26 decimal places. [TRANS.] 



24 

is an additive quantity; so that according to 
the classical theory it is meaningless to speak 
of energy at a certain point, since it is essential 
to state the physical system the energy of which 
is tinder discussion; just as it is similarly impos- 
sible to speak of a definite velocity unless the 
system be indicated to which velocity is referred. 
Now we are at liberty to choose whatever physical 
system we please, either little or great; and 
consequently the value of the energy is always 
to a certain extent arbitrary. The difficulty, then, 
consists in the fact that this arbitrary energy is 
supposed to be equated with a localized fre- 
quency. The gulf between these two concepts 
should now be clearly apparent: and in order 
to bridge this gulf a step of fundamental import- 
ance must be taken. This step does imply a 
break with those assumptions which classical 
Physics has always regarded and employed as 
axiomatic. 

Hitherto it had been believed that the only 
kind of causality with which any system of 
Physics could operate was one in which all the 
events of the physical world by which, as usual, 
I mean not the real world but the world-view of 
Physics might be explained as being composed 
of local events taking place in a number of 



individual and infinitely small parts of Space. 
It was further believed that each of these ele- 
mentary events was completely determined by 
a set of laws without respect to the other events; 
and was determined exclusively by the local 
events in its immediate temporal and spatial 
vicinity. Let us take a concrete instance of 
sufficiently general application. We will assume 
that the physical system under consideration 
consists of a system of particles, moving in a 
conservative field of force of constant total 
energy. Then according to classical Physics each 
individual particle at any time is in a definite 
state; that is, it has a definite position and a 
definite velocity, and its movement can be 
calculated with perfect exactness from its initial 
state and from the local properties of the field 
of force in those parts of Space through which 
the particle passes in the course of its movement. 
If these data are known, we need know nothing 
else about the remaining properties of the system 
of particles under consideration. 

In modern mechanics matters are wholly 
different. According to modern mechanics, merely 
local relations are no more sufficient for the 
formulation of the law of motion than would 
be the microscopic investigation of the different 



26 

parts of a picture in order to make clear its 
meaning. On the contrary, it is impossible to 
obtain an adequate version of the laws for which 
we are looking, unless the physical system is 
regarded as a Whole. According to modern 
mechanics, each individual particle of the system, 
in a certain sense, at any one time, exists simul- 
taneously in every part of the space occupied by 
the system. This simultaneous existence applies 
not merely to the field of force with which it is 
surrounded, but also to its mass and its charge. 

Thus we see that nothing less is at stake here 
than the concept of the particle the most ele- 
mentary concept of classical mechanics. We are 
compelled to give up the earlier essential meaning 
of this idea; only in a number of special border- 
line cases can we retain it. But if we pursue the 
line of thought indicated above, we shall find 
what it is that we can substitute for the concept 
of the particle in more general cases. 

[The following brief section may be omitted by readers 
not interested in the somewhat technical issues, and the 
subject resumed on p. 38.] 

[The Quantum Theory postulates that an 
equation subsists between energy and frequency. 
If this postulate is to have an unambiguous 
meaning, that is a meaning independent of the 



27 

particular system to which it is referred, then 
the Principle of Relativity demands that a 
momentum vector 1 shall be equivalent to a wave- 
member vector; in other words, the absolute 
quantity of the momentum must be equivalent 
to the reciprocal of the length of a wave whose 
normal coincides with the direction of momentum. 
The wave in question must not be imagined as 
existing in ordinary three-dimensional space, but 
in so-called configuration space, the dimension 
of which is given by the number of degrees of 
freedom of the system, and in which the square 
of the element of length is measured by twice 
the kinetic energy; or what comes to the same 
thing, by the square of the total momentum. 
It thus appears that the wave-length follows 
from the kinetic energy, that is from the differ- 
ence between the constant total energy and the 
potential energy; this difference must be regarded 
as a function of position given beforehand. 

The product of the frequency and the wave- 
length gives us the rate of propagation of the 
wave; in other words, it gives us the phase- 
velocity of a given wave the so-called material 

1 A vector is a quantity which has a definite direction; 
for example, "100 miles per hour East" (or any other 
direction) is a vector. [TRANS.] 



28 

wave in configuration space. If the appropriate 
values are substituted in the familiar equation 
of classical mechanics, we obtain the linear 
homogeneous partial differential equation set 
up by Schrodinger. This equation has provided 
the basis of modern Quantum-mechanics, in 
which it seems to play the same part as do the 
equations established by Newton, Lagrange and 
Hamilton in classical mechanics. Nevertheless 
there is an important distinction between these 
equations, consisting in the fact that in the latter 
equations the co-ordinates of the configuration 
point are not functions of time, but independent 
variables. Accordingly, while for any given system 
the classical equations of motion were more or 
less numerous and corresponded to the number 
of degrees of freedom of the system, there can 
be only one single quantum-equation for each 
system. In course of time the configuration point 
of classical theory describes a definite curve; on 
the other hand, the configuration point of the 
material wave fills at any given time the whole 
of infinite space, including those parts of space 
where potential energy is greater than the total 
energy, so that according to the classical theory, 
kinetic energy would become negative in these 
parts of space, and the momentum imaginary. 



29 

This case resembles the so-called total reflection 
of light, where according to geometrical optics 
light is completely reflected, because the angle 
of refraction becomes imaginary; whereas accord- 
ing to the wave-theory of light, it is perfectly 
possible for light to penetrate into the second 
medium, even if it cannot do so as a plane wave. 

At the same time, the fact that there are 
points in configuration space where the potential 
energy exceeds the total energy is of extreme 
impoitance for Quantum-mechanics. Calculation 
shows that in every such instance a finite wave 
corresponds not to any given value of the energy 
constant, but corresponds only to certain definite 
values : the so-called characteristic energy- values, 
which can be calculated from the wave-equation 
and have different values according to the nature 
of the given potential energy. 

From the discrete characteristic energy- values, 
discrete characteristic values of the period of 
oscillation may be derived. The latter are deter- 
mined according to the Quantum postulate, in 
a similar manner to that of a stretched cord 
with fixed ends; with this distinction that the 
latter quantization is determined by an external 
condition, viz. the length of the cord, whereas 
in the present instance it depends upon the 



30 

Quantum of Action, which in turn depends 
directly upon the differential equation. 

To each characteristic vibration there corre- 
sponds a particular wave-function (0); this is 
the solution of the wave-equation; and all these 
different characteristic functions form the com- 
ponent elements for the description of any 
movement in terms of wave-mechanics. 

Thus we reach the following result : in classical 
Physics the physical system under consideration 
is divided spatially into a number of smallest 
parts; by this means the motion of material 
bodies is traced back to the motion of their com- 
ponent particles, the latter being assumed to 
be unchangeable. In other words, the explana- 
tion is based upon a theory of corpuscles. Quan- 
tum Physics, on the other hand, analyses all 
motion into individual and periodic material 
waves, which are taken to correspond to the 
characteristic vibrations and characteristic func- 
tions of the system in question; in this way it 
is based upon wave-mechanics. Accordingly, in 
classical mechanics the simplest motion is that 
of an individual particle, whereas in quantum- 
mechanics the simplest motion is that of a 
simple periodic wave; according to the first, 
the entire motion of a body is taken as being the 



31 

totality of the motions of its component particles; 
whereas according to the second, it consists in 
the joint effect of all kinds of periodic material 
waves. To illustrate the difference between these 
two views, we may once more refer to the 
vibrations of a stretched cord. On the one hand 
these vibrations may be imagined as consisting 
of the sum of the motions of the different particles 
of the cord, where each particle is in motion 
independently of all the rest and in accordance 
with the force acting upon it, which in turn 
depends upon the local curvature of the cord. 
On the other hand the process of vibration may 
be analysed into the fundamental and upper 
partial vibrations of the cord, where each vibration 
affects the cord in its totality and the sum total 
of vibration is the most general kind of motion 
taking place in the cord. 

Wave-mechanics also furnishes an explanation 
for another fact which hitherto has been inex- 
plicable. According to Niels Bohr's theory, the 
electrons of an atom move around the nucleus 
in accordance with laws very similar to those 
which govern the motion of the planets around 
the sun. Here the place of gravitation is taken 
by the attraction between the opposite charges 
of the nucleus and the electrons. There is, however, 



a curious distinction, consisting in the fact 
that the electrons can move only in definite 
orbits distinct from each other, whereas with 
the planets no one orbit appears to be privileged 
beyond any other. According to the wave theory 
of electrons this circumstance, at first sight unin- 
telligible, is easily explained. If the orbit of an 
electron returns upon itself, it is clear that it 
must comprise an integral number of wave- 
lengths, just as the length of a chain which forms 
a complete circle, if it consists of a number of 
equal links, must always equal an integral 
number of link-lengths. According to this view 
the revolution of an electron around the nucleus 
is not so much like the movement of a planet 
around the sun, as like the rotation of a sym- 
metrical ring upon its centre, so that the ring 
as a whole retains the same position in space; 
thus there is no physical meaning in referring 
to the local position of the electron at any 
instant. 

The following question may now be asked: if 
motion is to be analysed not into particles, but 
into material waves, what is the procedure of 
wave-mechanics when it is called upon to describe 
the motion of a single particle which occupies a 
given position at a given time? The answer to 



33 

this question throws light upon the great contrast 
between the two theories with which we have 
been dealing. In the first instance we must 
examine the physical meaning of the wave 
function ^ of a simple periodic material wave. 
This meaning can be derived from the considera- 
tion that the energy of a material wave has a 
twofold meaning. It is true that it denotes the 
period of vibration of the wave; but of course 
it does not follow from this that it has lost 
its original meaning, which it derives from the 
principle of conservation of energy. But if the 
energy principle is to apply to wave-mechanics, 
then it must be possible to represent the energy 
of a material wave, not only by the frequency of 
its vibrations, but also by means of an integral 
comprehending the entire configuration space of 
the wave. 

In fact, then, if the wave-equation is multiplied 
by ^ and the product is integrated over the 
entire configuration space, there results a definite 
expression for the energy, which can be most 
vividly interpreted in the following manner. 

We imagine the material system of particles 
under consideration to be multiplied many times, 
and we further imagine that each of the resulting 
systems is in a different configuration, so that 

Modern Physics _ 



34 

we obtain a very great number of particles in 
configuration space. We further allot to the con- 
figuration points existing in the different infinitely 
small elements of space a definite energy which 
is composed (a) of the value of the local potential 
energy (which is given beforehand) and (b) of a 
second element which varies as the square of the 
local gradient of iff, and which we can interpret 
as being equivalent to kinetic energy. If, then, 
the spatial density of the configuration points at 
any one place is assumed to be equal to the 
square of the absolute value of if/ (which latter 
we may assume to have any magnitude we 
desire, since one of the constant factors of iff 
can be selected by ourselves at will), it follows 
that the mean energy of all the configuration 
points is equivalent to the energy of the material 
wave. Accordingly the absolute value of the 
amplitude of the wave has no meaning whatever 
in a physical sense. If we imagine ^ to be selected 
in such a way that the square of the absolute 
value of 0, when integrated over the configura- 
tion space, gives us the value i, then we can 
also say that this square denotes the probability 
that the material system of particles is actually 
existing at the point in question within the 
configuration space. Thus we have found a vivid 



35 

expression for the physical meaning of the wave- 
function 0, which we were looking for. 

In the course of all these considerations we 
had assumed that $ had a definite characteristic 
function of its own, and that there was a simple 
periodic wave corresponding to it. Similar state- 
ments, however, may be made for the general 
case where waves having different periods are 
superimposed. In that case the wave-function */* 
is the algebraic sum of the periodic characteristic 
functions multiplied by a certain amplitude con- 
stant, and once again the square of the absolute 
value of iff denotes the probability for the corre- 
sponding position of the configuration point. In 
the general case, of course, we can no longer 
speak of one single definite period of vibration 
of the material waves; on the other hand, how- 
ever, we can still speak of a definite energy. 
Accordingly the Quantum-equation E = hv loses 
its original meaning and only gives us an average 
frequency v. It is worth noting here that if a 
sufficiently large number of different waves having 
approximately equal frequencies are superimposed, 
the wave-function of the resulting wave is the 
sum of the individual wave-functions ; its energy 
on the other hand does not increase proportion- 
ately with the number of individual waves, but 



-36- 

always retains its original mean value; the 
energy of a group of individual waves defines 
a mean frequency, and similarly the momentum 
of this group serves to define a mean wave-length. 
To begin with, the amplitudes and phases of 
the individual waves can be selected at will. 
Beyond this, however, it is impossible to intro- 
duce further variety into the mechanical processes 
of which wave-mechanics can provide instances. 
This fact becomes important when we turn to 
the question raised above, in which we ask how 
the motion of a single definite particle is to be 
described in terms of wave-mechanics. It appears 
immediately that such a description cannot be 
made in any exact sense. Wave-mechanics possesses 
only one means of defining the position of a 
particle, or more generally the position of a 
definite point in configuration space; this consists 
in superimposing a group of individual waves of 
the system, in such a manner that their wave- 
functions cancel each other by interference every- 
where within configuration space, and intensify 
each other only at the one point in question. In 
this case the probability of all the other con- 
figuration points would be equal to O, and would 
be equal to r only for the one point in question. 
In order to isolate this point completely we 



37 

should, however, require infinitely small wave- 
lengths, and consequently infinitely great momen- 
tum. Therefore, in order to obtain a result which 
would be even approximately useful, we should 
have to begin by substituting for the definite 
configuration point a finite (though still small) 
region of configuration space, or so-called wave- 
group; this sufficiently expresses the fact that 
ascertaining the position of a configuration point 
is always in the wave theory affected by some 
sort of uncertainty. 

If we wish to go further and ascribe to the 
system of particles a definite quantity of momen- 
tum as well as a definite configuration, then the 
Quantum postulate, if taken strictly, will allow 
us to make use of only one single wave of a 
definite length for our exposition, and once more 
description becomes impossible. On the other 
hand, if a slight uncertainty is allowed to creep 
into the quantity of momentum, then we can 
reach our goal, at least approximately, if we make 
use of the wave within a certain narrow range 
of frequency. 

According to wave-mechanics, both the posi- 
tion and the momentum of a sj ,cem of particles 
can never be defined without some uncer- 
tainty. Now the fact is that between these two 



_ 3 8- 

kinds of uncertainty there is a definite relation. 
This follows from the simple reflection that if 
the waves of which we make use are to cancel 
each other through interference outside the 
above-mentioned small configuration region, then 
in spite of their small difference in frequency, 
noticeable differences in propagation must appear 
at the opposite boundaries of the region. If in 
accordance with the Quantum postulate, we 
substitute differences of momentum for differ- 
ences of propagation, we obtain Heisenberg's 
Principle, which states that the product of the 
uncertainty of position and uncertainty of momen- 
tum is at least of the same order of magnitude 
as the quantum of action.] 

The more accurately the position of the con- 
figuration point is ascertained, the less accurate 
is the amount of momentum; and conversely. 
These two kinds of uncertainty are thus in a 
certain sense complementary; this complemen- 
tariness is limited by the fact that momentum 
can under certain conditions be defined with 
absolute accuracy in wave-mechanics, whereas 
the position of a configuration point always 
remains uncertain within a finite region. 

Now this relation of uncertainty, established 



39 

by Heisenberg, is something quite unheard of in 
classical mechanics. It had always been known, 
of course, that every measurement is subject to 
a certain amount of inaccuracy; but it had 
always been assumed that an improvement in 
method would lead to an improvement in 
accuracy, and that this process could be carried 
on indefinitely. According to Heisenberg, how- 
ever, there is a definite limit to the accuracy 
obtainable. What is most curious is that this 
limit does not affect position and velocity separ- 
ately, but only the two when combined together. 
In principle, either taken by itself can be measured 
with absolute accuracy, but only at the cost of 
the accuracy of the other. 

Strange as this assertion may seem, it is 
definitely established by a variety of facts. I 
will give one example to illustrate this. The 
most direct and accurate means of ascertaining 
the position of a particle consists in the optical 
method, when the particle is looked at with the 
naked eye or through a microscope, or else is 
photographed. Now for this purpose the particle 
in question must be illuminated. If this is done 
the definition becomes more accurate; conse- 
quently the measurement becomes more exact 
in proportion as the light-waves employed 



40 

become shorter and shorter. In this sense, then, 
any desired degree of accuracy can be attained. 
On the other hand there is also a disadvantage, 
which affects the measurement of velocity. Where 
the masses in question have a certain magnitude, 
the effect of light upon the illuminated object 
may be disregarded. But the case is altered if a 
very small mass, e.g. a single electron, is selected; 
because each ray of light, which strikes the 
electron and is reflected by it, gives it a distinct 
impulse ; and the shorter the light-wave the more 
powerful is this impulse. Consequently, the 
shorter the light-wave the more accurately is it 
possible to determine position; but at the same 
moment measurement of velocity becomes pro- 
portionately inaccurate ; and similarly in analogous 
instances. 

On the view which has just been set out classical 
mechanics, which is based on the assumption of 
unchanging and accurately measurable corpuscles 
moving with a definite velocity, forms one ideal 
limiting case. This ideal case is actually realized 
when the observed system possesses a relatively 
considerable energy. When this happens, the 
distinct characteristic energy values will lie close 
to each other, and a relatively small region of 
energy will contain a considerable number of 



4* 

high wave-frequencies (i.e. of short wave-lengths) ; 
through the superposition of these a small wave- 
group with definite momentum can be delimited 
comparatively accurately within the configuration 
space. In this case, wave-mechanics merges with 
the mechanics of particles; Schrodinger's differ- 
ential equation becomes the classical differential 
equation of Hamilton and Jacobi, and the wave- 
group travels in configuration space in accor- 
dance with the same laws which govern the motion 
of a system of particles according to classical 
mechanics. But this state of affairs is of a limited 
duration; for the individual material waves are 
not interfering continually in the same manner, 
and consequently the wave-group will disintegrate 
more or less quickly; the position of the rela- 
tive configuration point will become more and 
more uncertain, and finally the only quantity 
remaining that is accurately defined is the wave- 
function tff. 

The question now arises whether these con- 
clusions correspond with experience. Since the 
Quantum of Action is so small, this question can 
be answered only within the framework of atomic 
physics; consequently the methods employed will 
always be extremely delicate. At present we can 
only say that hitherto no fact has been discovered 



42 

which throws doubt on the applicability in Physics 
of all these conclusions. 

The fact is that since the wave-equation was 
first formulated, the theory has been developing 
at a most remarkable rate. It is impossible within 
the framework of a small volume to mention all 
the extensions and applications of the theory 
which have been evolved within recent years. I 
shall confine myself to the so-called stress of 
protons and electrons; the formulation of Quan- 
tum-mechanics in terms of Relativity; the 
application of the theory to molecular problems, 
and the treatment of the so-called "many-body 
problem", i.e. its application to a system con- 
taining a number of exactly similar particles. 
Here statistical questions, relating to the number 
of possible states within a system, having a given 
energy, are particularly important; they also 
have a bearing on the calculation of the entropy 
of the system. 

Finally, I cannot here enter in detail upon the 
Physics of light-quanta. In a certain sense this 
study has developed in the opposite direction 
from the Physics of particles. Originally Maxwell's 
theory of electromagnetic waves dominated this 
region, and it was not seen until later that we 
must assume the existence of discrete light- 



43 

particles; in other words that the electromagnetic 
waves, like the material waves, must be inter- 
preted as waves of probability. 

Perhaps there is no more impressive proof of 
the fact that a pure wave theory cannot satisfy 
the demands of modern Physics any more than 
a pure corpuscular theory. Both theories, in fact, 
represent extreme limiting cases. The corpuscular 
theory, which is the basis of classical mechanics, 
does justice to the configuration of a system, but 
fails to determine the values of its energy and of 
momentum; conversely the wave theory, which 
is characteristic of classical electrodynamics, can 
give an account of energy and momentum, but 
excludes the idea of the localization of light- 
particles. The standard case is represented by 
the intermediate region, where both theories play 
equally important paits; this region can be 
approached from either side, although at present 
a close approach is impossible. Here many 
obscure points await solution, and it remains to 
be seen which of the various methods employed 
for their solution best leads to the goal. Among 
them we may mention the matrix calculus invented 
by Heisenberg, Born, and Jordan, the wave theory 
due to de Broglie and Schrodinger, and the mathe- 
matics of the q numbers introduced by Dirac. 



44 

4 

If we attempt to draw a comprehensive conclu- 
sion from the above description and to obtain an 
insight into the distinguishing characteristics of 
our new picture of the world, the first impression 
will no doubt be somewhat unsatisfactory. First 
of all it will appear surprising that wave-mechanics, 
which itself is in complete contradiction to classi- 
cal mechanics, nevertheless makes use of concepts 
drawn from the classical corpuscular theory; e.g. 
the concept of the co-ordinates and momentum 
of a particle, and of the kinetic and potential 
energy of a system of particles. The contradiction 
is the more surprising since it afterwards proved 
impossible simultaneously to determine exactly 
the position and momentum of a particle. At the 
same time these concepts are absolutely essential 
to wave-mechanics; for without them it would 
be impossible to define configuration space and 
ascertain its measurements. 

There is another difficulty attached to the wave 
theory, consisting in the fact that material waves 
are not as easy to bring before the imagination 
as are acoustic or electromagnetic waves; for 
they exist in configuration space instead of 
ordinary space, and their period of vibration 



45 

depends on the choice of the physical system to 
which they belong. The more extensive this 
system is assumed to be, the greater will be its 
energy, and with this the frequency. 

It must be admitted that these are serious 
difficulties. It will be possible, however, to over- 
come them if two conditions are fulfilled: the 
new theory must be fiee from internal contra- 
dictions; and its applied results must be definite 
and of some significance for measurement. At the 
present time opinions are somewhat divided 
whether these requirements are fulfilled by 
Quantum-mechanics, and if so, to what extent. 
For this reason I propose to discuss this funda- 
mental point further. 

It has frequently been pointed out that Quan- 
tum-mechanics confines itself on principle to 
magnitudes and quantities which can be observed, 
and to questions which have a meaning within 
the sphere of Physics. This obseivation is correct; 
but in itself it must not be considered a special 
advantage of the Quantum Theory as opposed 
to other theories. For the question whether a 
physical magnitude can in principle be observed, 
or whether a certain question has a meaning as 
applied to Physics, can never be answered a 
priori, but only from the standpoint of a given 



- 4 6~ 

theory. The distinction between the different 
theories consists precisely in the fact that 
according to one theory a certain magnitude can 
in principle be observed, and a certain question 
have a meaning as applied to Physics; while 
according to the other theory this is not the 
case. For example, according to the theories of 
Fresnel and Lorentz, with their assumption of 
a stationary ether, the absolute velocity of the 
earth can in principle be observed; but according 
to the Theory of Relativity it cannot ; again, the 
absolute acceleration of a body can be in prin- 
ciple observed according to Newtonian mechanics, 
but according to Relativity mechanics it cannot. 
Similarly the problem of the construction of a 
perpetuum mobile had a meaning before the 
principle of the conservation of energy was 
introduced, but ceased to have a meaning after 
its introduction. The choice between these two 
opposed theories depends not upon the nature 
of the theories in themselves, but upon experi- 
ence. Hence it is not sufficient to describe the 
superiority of Quantum-mechanics, as opposed 
to classical mechanics, by saying that it confines 
itself to quantities and magnitudes which can in 
principle be observed, for in its own way this is 
true also of classical mechanics. We must indicate 



47 

the particular magnitudes or quantities which, 
according to Quantum-mechanics, are or are not 
in principle observed; after this has been done 
it remains to demonstrate that experience agrees 
with the assertion. 

Now this demonstration has in fact been com- 
pleted, e.g. with respect to Heisenberg's Principle 
of Uncertainty, so far as seems possible at the 
present moment, and to this extent it can be 
looked upon as proving the superiority of wave- 
mechanics. 

In spite of these considerable successes, the 
Principle of Uncertainty which is characteristic 
of Quantum Physics has caused considerable 
hesitation, because the definition of magnitudes 
and quantities which are continually in use is 
in principle treated as being inexact by this 
theory. This dissatisfaction is increased by the 
fact that the concept of probability has been 
introduced in the interpretation of the equations 
used in Quantum-mechanics; for this seems to 
imply a surrender of the demands of strict 
causality in favour of a form of indeterminism. 
To-day, indeed, there are eminent physicists who 
under the compulsion of facts are inclined to 
sacrifice the principle of strict causality in the 
physical view of the world 



- 4 8- 

If such a step should actually prove necessary 
the goal of physicists would become more remote; 
and this would be a disadvantage whose impor- 
tance it is impossible to overestimate. For in my 
opinion, so long as any choice remains, deter- 
minism is in all circumstances preferable to 
indeterminism, simply because a definite answer 
to a question is always preferable to an indefinite 
one. 

So far as I can see, however, there is no ground 
for such a renunciation. For there always remains 
the possibility that the reason why it is impos- 
sible to give a definite answer resides, not in the 
nature of the theory, but in the manner in which 
the question is asked. If a question is inadequately 
formulated physically, the most perfect physical 
theory can give no definite answer; a fact widely 
known in classical statistics and frequently dis- 
cussed. For example, if two elastic spheres strike 
one another in a plane, while their velocities 
before impact and the laws of impact are known 
in all their details, it still remains impossible to 
state their velocities after impact. The fact is 
that, in order to calculate the four unknown 
components of the velocities of the two spheres 
after impact, we have only three equations 
derived from the conservation of energy and the 



49 

two components of momentum. From this, how- 
ever, we do not infer that there is no causality 
governing impact phenomena; what we do say 
is that certain essential data axe missing which 
are requisite for their complete determination. 

In order to apply these considerations to the 
problems of Quantum Physics, we must now 
return to the arguments dealt with in the Intro- 
duction. 

If it is really true that, in its perpetual changes, 
the structure of the physical world-view moves 
further and further away from the world of the 
senses, and correspondingly approaches the real 
world (which, as we saw, cannot in principle be 
apprehended at all), then it plainly follows that 
our view of the world must be purged progres- 
sively of all anthropomorphic elements. Conse- 
quently we have no right to admit into the 
physical world-view any concepts based in any 
way upon human mensuration. In fact this is 
not the case with Heisenberg's Principle of 
Uncertainty: this was reached from the con- 
sideration that the elements of the new view of 
the world are not material corpuscles, but simple 
periodic material waves which correspond to the 
physical system under consideration a conclu- 
sion obtained in accordance with the mathematical 

Modern Physics _ 



50 

principle that it is impossible to determine a 
definite particle with definite momentum by 
means of superposition of simple periodic waves 
having a finite length. The principle has nothing 
whatever to do with any measurement, while 
the material waves are definitely determined by 
means of the mathematical problem of boundary 
values relating to the case in question. Here 
there is no question of indeterminism. 

The question of the relation between the 
material waves and the world of sense is a 
different one. For this relation renders it possible 
for us to become acquainted with physical 
events ; if a system were completely cut off from 
its surroundings we could never know of its 
existence. 

At first glance it appears that this question 
has nothing to do with Physics, since it belongs 
partly to Physiology and partly to Psychology. 
These objections, however, lead to no real diffi- 
culty. It is always possible to imagine suitably 
constructed instruments being substituted for 
human senses, e.g. self-registering apparatus like 
a sensitive film, which registers the impressions 
derived from the environment, and is thus capable 
of furnishing evidence about the events taking 
place in these surroundings. If such instruments 



are included within the physical system which 
we propose to consider, and if all other influences 
are eliminated, then we have a physical system 
cut off from the rest of the world of which we 
can discover something by means of measure- 
ment ; although it is true that we must take into 
account the structure of the measuring instru- 
ments, and the reaction which they might con- 
ceivably have upon the events which we desire 
to measure. 

If we possessed an instrument reacting to a 
simple periodic material wave in the same way 
as a resonator reacts to a sound-wave, then we 
would be in a position to measure individual 
material waves and thus to analyse their behaviour. 
This is not the case; the fact is that the indica- 
tions given by such instruments as we possess, 
e.g. the darkening of a photographic film, do not 
allow us to make a safe inference about all the 
details of the process under examination. We 
have no right, however, to infer from this that 
the laws of material waves are indeterminate. 

Another and more direct attempt might be 
made to substantiate the assumption of indeter- 
minism from the fact that, according to wave- 
mechanics, the events within a system of particles 
cut off from the outside world are not determined 



in any way by the initial state of the system, 
i.e. by the initial configuration and initial momen- 
tum. There is not even an approximate deter- 
mination; for the wave-group corresponding to 
the initial state will in time disintegrate generally 
and fall apart into individual waves of probability. 

On closer consideration, however, we see that 
in this instance the element of indeterminism is 
due to the manner in which the question is asked. 
The question is based upon corpuscular mechanics; 
and in corpuscular mechanics the initial state 
governs the course of the event for all time. 
But in wave-mechanics such a question has no 
place, if only because the final result is on prin- 
ciple affected with a finite inaccuracy due to the 
Principle of Uncertainty. 

Since the times of Leibniz, on the other hand, 
another form of question in classical mechanics 
has been known which in this sphere leads to a 
definite answer. An event is completely deter- 
mined for all time if, apart from the configuration 
at a certain time, we know, not the momentum, 
but the configuration of the same system at a 
different instant. In this case the principle of 
variation, or principle of least action, is used in 
order to calculate the event. To take the previous 
example, where two elastic spheres meet in a 



53 

plane, if we know the initial and final position 
of the spheres and the interval between those 
two positions, then the three unknown quantities, 
namely the two local co-ordinates and the time 
co-ordinate of impact, are completely determined 
by the three equations of conservation. 

This changed formulation of the problem 
differs from the previous formulation because it 
is immediately applicable to wave-mechanics. 
It is true, as we saw, that a given configuration 
can never be defined with complete accuracy 
by the wave theory; but on the other hand it 
is theoretically possible to reduce the uncertainty 
below any desired limit, and thus to determine 
the event in question with any desired degree 
of accuracy. Further, the disintegration of wave- 
groups is no evidence in favour of indeterminism, 
since it is equally possible for a wave-group to 
conglomerate: in both the wave theory and the 
corpuscular theory the direction of the process 
is immaterial. Any movement might equally well 
take place in the opposite direction. 

When the above formulation of the problem 
is adopted a given wave-group generally, of 
course, exists only at the two selected instants: 
in the intervening period, as well as before and 
after the process, the different elementary waves 



54 

will exist separately. But whether they are 
described as material waves or as waves of prob- 
ability, in either case they will be completely 
determined. This is the explanation of the apparent 
paradox, that when a physical system passes by 
a definite process from one definite configuration 
during a definite time into some other definite 
configuration, the question what its configurations 
are during the intervening period has no physical 
significance; similarly on this view there is no 
meaning in the question of what is the track of 
light quantum emitted from a point source and 
absorbed at a given point on an observation- 
screen. 

It should at the same time be emphasized 
that on this view the meaning of determinism 
is not exactly what it is in classical Physics. In 
the latter the configuration is determined; in 
Quantum Physics, the material waves. The dis- 
tinction is important, because the connection 
between the configuration and the world of sense 
is far more direct than that between the material 
waves and the sense-world. To this extent the 
relation between the physical world-view and 
the world of sense appears to be considerably 
looser in modern Physics. 

This is undoubtedly a disadvantage; but it 



e> gt ... 

35 

is the price that must be paid in order to pre- 
serve the determinism of our world-view. And 
further, this step appears to lie in the general 
direction in which Physics is actually developing; 
this has been pointed out on more than one 
occasion, since in the course of its progressive 
evolution, the structure of the physical view of 
the world is moving farther and farther away 
from the world of sense, and assuming more and 
more abstract forms. Indeed, the principle of 
Relativity seems actually to demand such a 
view; for on this principle Time stands on the 
same level with Space, whence it follows that, if 
a finite space is required for the causal description 
of a physical process, a finite temporal interval 
must also be used in order to complete the 
description. 

On the other hand, it may well be that the 
suggested formulation of the question is too 
one-sided, and too anthropomorphic to furnish 
satisfactory material for a new theory of the 
structure of the physical world; it may be that 
we shall have to look for some other formulation. 
In any case many complex problems remain to 
be solved, and many obscure points to be 
cleared up. 

In view of the peculiar difficulties of the 



position which has been reached by theoretical 
Physics, a feeling of doubt persists whether the 
theory, with all its radical innovations, is really 
on the right path. The answer to this decisive 
question depends wholly upon the degree of 
necessary contact with the sense world which 
the physical world-view maintains in the course 
of its incessant advance. If this contact is lost 
even the most perfect world-view would be no 
better than a bubble ready to burst at the first 
puff of wind. There is, fortunately, no cause for 
apprehension, at least in this respect: indeed we 
may assert without exaggeration that there was 
no period in the history of Physics when theory 
and experience were linked so closely together 
as they are now. Conversely, it was the facts 
learned from experiments that shook and finally 
overthrew the classical theory. Each new idea 
and each new step were suggested to investi- 
gators, where it was not actually thrust upon 
them, as the result of measurements. The Theory 
of Relativity was led up to by Michelson's experi- 
ments on optical interference, and the Quantum 
Theory by Lummer's, Pringsheim's, Ruben's and 
Kurlbaum's measurements of the spectral distri- 
bution of energy, by Lenard's experiments on 
the photoelectric effect, and by Franck and 



57 """" 

Hertz's experiments on the impact of electrons. 
It would lead me too far if I were to enter on 
the numerous and surprising results which have 
compelled Physical theory to abandon the 
classical standpoint and to enter on a definite 
new course. 

We can only hope that no change will take 
place in this peaceful international collaboration. 
It is in this reciprocal action of experiment and 
theory which is at once a stimulus to and a 
check upon progress that we see the surest and 
indeed the only guarantee of the future advance 
of Physics. 

What will be the ultimate goal ? I had occasion 
at the beginning to point out that research in 
general has a twofold aim the effective domina- 
tion of the world of sense, and the complete under- 
standing of the real world; and that both these 
aims are in principle unattainable. But it would 
be a mistake to be discouraged on this account. 
Both our theoretical and practical tangible results 
are too great to warrant discouragement; and 
every day adds to them. Indeed, there is perhaps 
some justification for seeing in the very fact 
that this goal is unattainable, and the struggle 
unending, a blessing for the human mind in its 
search after knowledge, For it is in this way that 



58- 

its two noblest impulses enthusiasm and rever- 
ence are preserved and inspired anew. 



5 



What now do we mean by physical law? A 
physical law is any proposition enunciating a 
fixed and absolutely valid connection between 
measurable physical quantities a connection 
which permits us to calculate one of these quan- 
tities if the others have been discovered by 
measurement. The highest and most keenly 
desired aim of any physicist is to obtain the 
most perfect possible knowledge of the laws of 
Physics, whether he looks at them from a utili- 
tarian point of view and values them because 
they enable him to save himself the trouble of 
costly measurements, or takes a deeper view and 
looks to them for satisfaction of a profound 
yearning after knowledge and for a firm basis 
of natural science. 

How do we discover the individual laws of 
Physics, and what is their nature? It should be 
remarked, to begin with, that we have no right 
to assume that any physical laws exist, or if they 
have existed up to now, that they will continue 



59 ~ 

to exist in a similar manner in future. It is per- 
fectly conceivable that one fine day Nature 
should cause an unexpected event to occur which 
would baffle us all; and if this were to happen 
we would be powerless to make any objection, 
even if the result would be that, in spite of our 
endeavours, we should fail to introduce order 
into the resulting confusion. In such an event, 
the only course open to science would be to 
declare itself bankrupt. For this reason, science 
is compelled to begin by the general assumption 
that a general rule of law dominates throughout 
Nature, or, in Kantian terminology, to treat the 
concept of causality as being one of the categories 
which are given a priori and without which no 
kind of knowledge can be attained. 

From this it follows that the nature of the laws 
of Physics, and the content of these laws, cannot 
be obtained by pure thought; the only possible 
method is to turn to the investigation of Nature, 
to collect the greatest possible mass of varied 
experiences, to compare these and to generalize 
them in the simplest and most comprehensive 
proposition. In other words, we must have 
recourse to the method of induction. 

The content of an experience is proportionally 
richer as the measurements upon which it is 



60 

based are more exact. Hence it is obvious that 
the advance of physical knowledge is closely 
bound up with the accuracy of physical instru- 
ments and with the technique of measurement. 
The latest developments of Physics provide us 
with striking examples of the truth of this. 
Measurement alone, however, does not suffice. 
For each measurement is an individual event 
standing by itself; as such, it is determined by 
special circumstances, especially by a definite 
place and a definite time, but also by a definite 
measuring instrument, and by a definite observer. 
It is true that frequently the generalization 
which is our object is quite obvious and, so to 
speak, thrusts itself upon us; on the other hand, 
there are cases where it is extremely difficult to 
find the common law governing a number of 
different measurements, either because it seems 
impossible to find such a law, or because a number 
of different laws seem available in order to 
generalize the facts. Both possibilities are equally 
unsatisfactory. 

In such cases, the only method of advance 
consists in introducing a so-called working hypo- 
thesis to see what it is worth and how far it will 
lead. It is generally a sign that the hypothesis 
is likely to turn out useful if it works even in 



6i 

those regions for which it was not originally 
designed. In such a case we have a right to 
assume that the law which it enunciates has a 
deeper meaning and opens the way to unmistak- 
ably new knowledge. 

We see then that a good working hypothesis 
is essential for inductive investigation. This 
being so, we are faced with the difficult question 
how we are to set about to find the most suitable 
hypothesis. For this there can be no general rule. 
Logical thought by itself does not suffice not 
even where it has an exceptionally large and 
manifold body of experience to aid it. The only 
possible method consists in immediately gripping 
the problem or in seizing upon some happy idea. 
Such an intellectual leap can be executed only 
by a lively and independent imagination and by 
a strong creative power, guided by an exact 
knowledge of the given facts so that it follows 
the right path. 

Such an intellectual process generally consists 
in the introduction of certain mental images and 
analogies which point the way to the reigning 
laws already known in other regions, thus sug- 
gesting a further step towards the simplification 
of the physical view of the world. 

It is precisely at these points where success 



62 

seems to be awaiting us, however, that a serious 
danger is frequently hidden. Once a step forward 
has succeeded and the working hypothesis has 
demonstrated its usefulness, we must go further. 
We have to reach the actual essence of the hypo- 
thesis and, by suitably formulating it, we have 
to throw light upon its genuine content by 
eliminating everything that is inessential. This 
process is not as simple as it might appear. The 
intellectual leap of which we spoke above con- 
structs a kind of bridge by which we can approach 
fresh knowledge; but on closer examination it 
frequently appears that this bridge is merely 
provisional, and that a more enduring structure 
must be put in its place, capable of bearing the 
heavy artillery of critical logic. 

We must bear in mind that every hypothesis 
is the outcome of the efforts of imagination, and 
that imagination works through direct intuition. 
But in Physics, as soon as we come to look for 
a rational theory or a logical demonstration, 
direct intuition is a very doubtful ally, however 
indispensable it may be while we are forming 
our hypothesis. For while it is natural that we 
should rely upon imaginations and ideas of this 
kind, which proved fruitful in one direction or 
another, such reliance is only too apt to lead to 



-6 3 ~ 

an overestimation of their importance and to 
untenable generalizations. We must further recog- 
nize that the authors of a new and practicable 
theory are frequently little inclined to introduce 
any important changes in the groups of ideas 
which led them towards their discoveries, whether 
from indolence or from a certain sentimental 
feeling, and that they often exert the whole of 
their well-earned authority in order to be able 
to maintain their original standpoint. Thus we 
shall readily understand the difficulties which 
often stand in the way of healthy theoretical 
development. Examples may be found at every 
point in the history of Physics, and I propose 
to enumerate some of the more important of 
them. 

The first exact measurements were made in 
the region of Space and Time the first region 
where accurate measurement was possible. Hence 
naturally the earliest physical laws were discovered 
in this field; in other words, in the sphere of 
mechanics. Again, we can readily understand 
how it came about that the first laws which were 
established related to those motions which occur 
regularly and independently of external inter- 
ference, namely, the motions of the celestial 
bodies. We know that the civilized peoples of the 



-6 4 - 

East had discovered thousands of years ago how 
to derive from their observations formulae which 
allowed them to calculate in advance the motion 
of the sun and the planets with great accuracy. 
Each improvement in the instruments of measure- 
ment was accompanied by an improvement of 
the formulae. By their co-ordination and com- 
parison the theories of Ptolemy, Copernicus and 
Kepler were evolved in course of time, each of 
which is simpler and more exact than those 
which preceded it. All these theories are alike 
in endeavouring to answer the question, what 
is the connection between the position of a 
celestial body, a planet for example, and the 
moment of time at which it occupies this position ? 
The nature of this necessary connection is, of 
course, different for the different planets, and 
this in spite of the fact that the motions of the 
planets have many characteristics in common. 

The decisive step beyond this type of question 
was taken by Newton. Newton summed up all 
the formulae relating to the planets in one single 
law governing their motion, and indeed that of 
all the celestial bodies. He was enabled to do 
this because he made the law of motion inde- 
pendent of the particular moment to which it 
is applied: for the instant He substituted the 



- 6 5 - 

time-differential. Newton's theory of planetary 
motion enunciates a fixed connection not between 
the position of a planet and time, but between 
the acceleration of a planet and its distance from 
the sun. Now this law a vectorial differential 
equation is the same for all the planets. Hence if 
the position and velocity of a planet are known 
for any moment, then its motion for all time 
can be exactly calculated. 

The successes obtained as the result of the 
further application of Newton's formulation of 
the laws of motion prove that it is not merely 
a new description of certain natural phenomena, 
but that it represents a real advance in the 
understanding of actual facts. It is not merely 
more exact than Kepler's formulae, for example 
when it allows for the interference in the elliptical 
orbit of the earth around the sun due to the 
periodic proximity of Jupiter, where formula and 
measurement are in exact agreement; more than 
that, it also covers the motion of such bodies as 
comets, twin stars, etc., which altogether elude 
Kepler's laws. The complete and immediate suc- 
cess of Newton's theory was due, however, to 
the fact that when applied to motion occurring 
on the earth, it led to the same numerical laws 
of gravitation and pendulum movements which 

Modern Physics -. 



66 

Galileo had already discovered by measurement, 
and also threw light on otherwise inexplicable 
phenomena, such as those of tides, rotation of 
the plane of the pendulum, precession of the axis 
of rotation, etc. 

The question which especially interests us at 
the moment is how Newton reached his differ- 
ential equation for planetary motion. He did 
not reach it by establishing a connection between 
the acceleration of a planet and its distance from 
the sun, and by looking for a numerical connec- 
tion between them; what he did was first to 
forge an intellectual link between them, leading 
from the concept of the position of a planet to 
that of its acceleration; and this link he called 
Force. He assumed that the position of a planet 
relatively to the sun depends upon a force of 
attraction directed towards the sun, and that 
the same attractive force also causes a definite 
change in the planet's motion. This was the 
germ of the law of gravitation, as well as of the 
law of inertia. The notion of force was no doubt 
derived (as the word implies) from the idea of 
the muscular sensation which arises when a 
weight is lifted or a ball is thrown; this idea 
was generalized and applied to every kind of 
change of motion, even where the forces in ques- 



-6 7 - 

tion are so great that no human power could 
possibly suffice to effect them. 

Small wonder, then, that Newton attributed 
the greatest importance to the concept of force 
which had helped him to reach such striking 
results. At the same time it must be noted that 
this concept does not occur in the law of motion 
proper. Newton looked to the concept of force 
for an explanation of every change of motion; 
and thus it came about that Newtonian force 
was regarded as the main and fundamental con- 
cept in mechanics, and not only in mechanics, 
but also in Physics; so that, in course of time, 
physicists formed the habit of making their first 
question when dealing with physical phenomena: 
what force is here in action ? 

Recent developments in Physics present a 
certain contrast with Newton's theory, so that 
in a manner it is true to say that the concept 
of force is no longer of fundamental importance 
for physical theory. In modern mechanics force 
is no more than a magnitude of secondary import- 
ance, and its place has been taken by higher and 
more comprehensive concepts that of work 1 or 

i "Work" is here used in its scientific sense of the pro- 
duct of the force and the distance through which the 
force acts. [TRANS,] 



68 

potential, where force in general is defined as a 
negative potential gradient. 

It might here be objected that work surely 
cannot be looked upon as something primary, 
since there must be some kind of force in exist- 
ence that does the work. This kind of argument 
is of the physiological and not of the physical 
order. It is true that in lifting a weight the con- 
traction of the muscles and the accompanying 
sensations are primary, and are the cause of the 
motion which actually takes place. But this kind 
of work, which is a physiological process, must 
be clearly distinguished from the physical force 
of attraction with which alone we are here con- 
cerned; it is this force which the earth exerts 
upon everything having weight; and this in its 
turn depends upon the gravitational potential 
which is already in existence and is primary. 

The idea of potential is superior to that of 
force, partly because it simplifies the laws of 
Physics, and also because the significance of the 
idea of potential has a far greater scope than 
that of force; it reaches beyond the sphere of 
mechanics into that of chemical affinities, where 
we are no longer concerned with Newtonian force. 
It must be admitted that the idea of potential 
has not the advantage of immediate obviousness 



-6 9 - 

which belongs to force by virtue of its anthropo- 
morphic quality; whence it follows that the 
elimination of the concept of force renders the 
laws of Physics much less obvious and easy of 
understanding. Yet this development is quite 
natural; the laws of Physics have no considera- 
tion for the human senses; they depend upon 
facts, and not upon the obviousness of facts. 

In my opinion, the teaching of mechanics will 
still have to begin with Newtonian force, just as 
optics begins with the sensation of colour, and 
thermodynamics with the sensation of warmth, 
despite the fact that a more precise basis is sub- 
stituted later on. Again, it must not be forgotten 
that the significance of all physical concepts and 
propositions ultimately does depend on their 
relation to the human senses. This is indeed 
characteristic of the peculiar methods employed 
in physical research. If we wish to form concepts 
and hypotheses applicable to Physics, we must 
begin by having recourse to our powers of 
imagination; and these depend upon our specific 
sensations, which are the only source of all our 
ideas. But to obtain physical laws we must 
abstract exhaustively from the images intro- 
duced, and remove from the definitions set up 
all irrelevant elements and all imagery which 



70 

do not stand in a logical connection with the 
measurements obtained. Once we have formu- 
lated physical laws, and reached definite con- 
clusions by mathematical processes, the results 
which we have obtained must be translated 
back into the language of the world of our senses 
if they are to be of any use to us. In a manner 
this method is circular; but it is essential, for 
the simplicity and universality of the laws of 
Physics are revealed only after all anthropo- 
morphic additions have been eliminated. 

The concept of Force as used by Newton is 
only one of a number of intellectual links and 
auxiliary notions employed in order to render 
an idea more intelligible. In this connection I 
should like to mention the idea of osmotic pressure 
introduced by van't Hoff. This idea proved 
particularly fruitful in physical chemistry, where 
it was used in order to formulate the physical 
laws of solutions, especially of the freezing-point 
and steam-pressure. To obtain instances of 
osmotic pressure, and measure it accurately, is 
not altogether easy, since an extremely complex 
apparatus (the so-called semi-permeable mem- 
branes) is required. We must the more admire 
the intuitive insight which led van't Hoff to 
formulate the laws known under his name despite 



the scantiness of the observed facts at his dis- 
posal. Yet in their present form these laws 
require osmotic pressure no more than the laws 
of motion require Newtonian Force. 

Besides the above there are other kinds of 
intellectual aids which assist imagination, and 
have proved of great assistance in the formation 
of working hypotheses, but which in the further 
course of development actually embarrassed later 
progress. One of these is particularly worth 
mention here. Men had accustomed themselves 
to see in some kind of force the cause underlying 
every natural change; and thus they were all 
the more disposed to imagine every invariable 
and constant magnitude or quantity as being 
of the nature of a Substance. From the earliest 
times the concept of Substance has played an 
important part in Physics: but closer examina- 
tion shows that this has not always been helpful. 
It is, of course, easy to see that wherever conser- 
vation is concerned, it is possible to assume a 
Substance of which conservation is predicated; 
and such an assumption undoubtedly makes it 
easier to grasp the meaning of the principle, and 
hence facilitates its use. A magnitude which in 
spite of every change retains its quantity surely 
cannot be imagined more vividly than in the 



72 

shape of a moving material body. It is a feature 
of this tendency that we are so prone to interpret 
all natural events as being movements of masses 
of substance a mechanistic interpretation. For 
example, the origin and distribution of light were 
explained by wave-motion in a substantial light 
ether; the chief laws of optics were described in 
this manner and found to agree with experience 
until the moment came when the mechanistic 
theory of Substance failed and became lost in 
unfruitful speculation. 

Again, for a time, the concept of Substance 
proved exceedingly useful as applied to Heat. 
The careful development of calorimetry, during 
the first half of the last century, was due in the 
main to the assumption that an unchanging 
heat-substance flowed from the warmer into the 
colder body. When it was shown that in these 
circumstances the amount of heat can be increased 
(e.g. by friction) the Substance theory defended 
itself by appealing to supplementary hypotheses. 
But although this method helped for some time, 
it did not avail indefinitely. 

In the theory of electricity the dangerous con- 
sequences of an exaggerated application of the 
idea of Substance became obvious at an early 
stage. Here again the idea of a subtle and quick- 



73 

moving electrical substance, giving rise to certain 
manifestations of force, serves admirably in order 
to render plastic before the mind such principles 
as that of the invariability of the quantity of 
electricity, and such subsidiary ideas as those of 
the electrical current and of the reciprocal action 
of charged conductors carrying a current. Here 
again, however, the analogy fails as soon as we 
have to allow for the fact that this view implies 
the assumption of the existence of two opposite 
substances, one positive and one negative, which 
completely neutralize each other when they are 
combined. Such an occurrence is at least as 
unthinkable as the creation of two opposite 
substances (in the usual sense) out of nothing. 

In this way we see that imaginative ideas and 
their resultant viewpoints must be used with 
the greatest caution, even when they have proved 
their value for some length of time, and despite 
the fact that they are indispensable for physical 
investigation and have provided the key to new 
knowledge on innumerable occasions. There is 
only one sure guide towards further development, 
and that is measurement, together with any 
logical conclusions that can be drawn from the 
concepts attached to this method. All other 
conclusions, and especially those characterized 



74 

by their so-called self-evidence, should always 
be looked upon with a certain suspicion. The 
validity of a proof dealing with well-defined 
concepts is to be judged by reason and not by 
intuition. 



6 



Up to this point we have been considering the 
manner in which the knowledge of physical laws 
is obtained. We will now proceed to examine the 
content and the essential nature of the laws of 
Physics in somewhat greater detail. 

A physical law is generally expressed in a 
mathematical formula, which permits us to 
calculate the temporal succession of the events 
taking place in a certain physical system under 
certain definite and given conditions. From this 
point of view all the laws of Physics can be divided 
into two main groups. 

The first group consists of those laws which 
remain valid even when the time order is reversed ; 
in other words, when every process that fulfils 
their requirements can take place in the reversed 
order without running counter to them. The laws 
of mechanics and of electrodynamics are of this 
nature, except in so far as they relate to chemical 



75 """" 

phenomena and the phenomena of heat. Every 
purely mechanical or electrodynamic process can 
take place in the reverse direction. The movement 
of a body falling without friction is accelerated 
in accordance with the same law which governs 
the retardation of a body rising without friction ; 
the same laws govern the movement of a pendu- 
lum to the left and to the right, and a wave can 
travel equally well in any direction and in any 
sense; a planet could equally well revolve around 
the sun westwards as eastwards. The question 
whether such movements could actually be 
reversed, and if so under what conditions, is 
another matter which need not here be discussed : 
we are now dealing with the law as such, not with 
the particular facts to which it applies. 

The laws belonging to the second group are 
characterized by the fact that their time order 
is of essential importance, so that the events 
taking place in accordance with these laws have 
only one temporal direction and cannot be 
reversed. Among these processes we may mention 
all those in which heat and chemical affinity play 
a part. Friction is always accompanied by a 
decrease and never by an increase of relative 
velocity; where heat is conducted the warmer 
body always becomes cooler and the cooler body 



- 7 6- 

warmer ; in diffusion the process invariably leads 
to a more thorough mixture and not to a pro- 
gressive separation of the substances in question. 
Further, these irreversible events always lead to 
a definite final state; friction to a relative state 
of rest, the transfer of heat to temperature 
equilibrium, and diffusion to a completely homo- 
geneous mixture. On the other hand the former 
class of reversible events knows neither beginning 
nor end, so long as no interference takes place 
from outside, but persists in incessant oscillation,, 
Now if we wish to introduce unity into the 
physical view of the universe we must somehow 
find a formula to cover both these contrasted 
types of law. How is this indispensable result 
to be brought about? Some thirty years ago 
theoretical physics was profoundly influenced by 
the so-called theory of Energetics, which sought 
to remove the antithesis by assuming that a fall 
in temperature, for example, was exactly analo- 
gous to the fall of a weight or of a pendulum 
from a higher to a lower position. This theory, 
however, did not take into consideration the 
essential fact that a weight can rise as well as 
fall, and that a pendulum has reached its greatest 
velocity at the moment when it has attained its 
lowest position and therefore, by virtue of its 



inertia, passes the position of equilibrium and 
moves to the other side. A transference of heat 
from a warmer to a colder body, on the contrary, 
diminishes with the diminution of the difference 
in temperature, while, of course, there is no such 
thing as any passing beyond the state of tem- 
perature equilibrium by reason of some kind of 
inertia. 

In whatever way we look at it, the contrast 
between reversible and irreversible processes 
persists; it must therefore be our task to find 
some entirely new point of view which will allow 
us to see that after all there is some connection 
between the different types of laws. Perhaps we 
shall succeed in showing that one group of laws 
is a derivative of the other; if so, the question 
arises which is to be considered the more simple 
and elementary the reversible processes or the 
irreversible. 

Some light is thrown on this question by a 
formal consideration. Every physical formula 
contains a number of constant magnitudes, 
together with variable magnitudes which have 
to be determined by measurement from case to 
case. The former magnitudes are fixed once for 
all and give its characteristic form to the func- 
tional connection between the variables which 



- 7 8- 

is expressed in the formula. Now if we examine 
these constants more carefully, we shall find 
that they invariably are the same for the rever- 
sible processes, always recurring, however widely 
different are the attendant outer conditions. 
Among these are mass, the gravitation constant, 
the electrical charge and the velocity of light. 
On the other hand the constants of the irre- 
versible processes, like the capacity for con- 
ducting heat, the coefficient of friction and the 
diffusion constant, depend to a greater or less 
degree on external circumstance, e.g. temperature, 
pressure, etc. 

These facts naturally lead us to regard the 
constants of the first group as the simpler, and 
the laws dependent on them as the more elemen- 
tary, and to suppose them incapable of further 
analysis, while treating the constants of the 
second group, and the laws depending on them, 
as being of a somewhat more complex nature. 
In order to test the validity of this assumption 
we must make our method of investigation some- 
what more exact; we must, so to speak, apply 
a lens of greater power to the phenomena. If the 
irreversible processes are in fact composite, then 
the laws governing them can only be roughly 
valid, so to say; they must be of a statistical 



79 

nature, since they are valid only for a large scale 
view or for summary consideration; that is, for 
the average values resulting from a large number 
of distinct processes. The more we restrict the 
number of individual events on which these 
average values are based, the more plainly will 
occasional divergences from the general or macro- 
scopic law make themselves felt. In other words, 
if in fact the view described is correct, then the 
laws of the irreversible processes, like those of 
friction, heat distribution and diffusion, must 
without exception be inexact if looked at micro- 
scopically; they must admit of exceptions in 
individual cases; and these exceptions will be 
the more striking, the more careful our examina- 
tion becomes. 

Now it so turned out in the course of events 
that experience tended more and more to confirm 
this conclusion. This could come about, of course, 
only as the result of a great improvement in the 
methods of making measurements. The laws 
governing the irreversible processes come so very 
near to being absolutely valid because of the 
enormous number of individual events of which 
these processes are composed. If, for example, 
we take a liquid having the same uniform tem- 
perature throughout, then it follows by the 



8o 

general or macroscopic law of the conduction of 
heat that no heat flows within the liquid. Such 
however is not precisely the case. For heat is 
the result of slight and rapid movements of the 
molecules constituting the liquid ; the conduction 
of heat, consequently, is due to the transference 
of these velocities when the molecules collide. 
Hence a uniform temperature does not mean 
that all the velocities are equal, but that the 
average value of the velocities for each small 
quantity of liquid is equal. This quantity in fact 
comprises a large number of molecules. But if 
we take a quantity containing a relatively small 
number of molecules, then the average of their 
velocities will vary; and the variation will be 
the greater, the smaller is the quantity of liquid. 
This principle can nowadays be regarded as a 
fact fully proved by experiment. One of the 
most striking illustrations is what is known as 
the Brownian Movement, which can be observed 
through the microscope in small particles of 
powder suspended in liquid. These particles are 
driven backwards and forwards by the invisible 
molecules of the liquid; the movement is the 
more pronounced the higher is the temperature. 
If we make the further assumption, to which in 
principle there is no objection, that each indivi- 



8i 

dual impulse is a reversible event governed by 
the strict elementary laws of dynamics, then we 
may say that the introduction of a microscopic 
method of examination shows that the laws 
governing the irreversible processes, or what is 
the same thing, the laws based upon statistics 
and mere rough approximation, can be traced 
back to dynamic, accurate, and absolute laws. 

The striking results reached by the introduction 
of statistical laws in many branches of physical 
research in recent times have produced a reftiark- 
able change in the views of physicists. They no 
longer, as in the earlier days of Energetics, deny 
or attempt to cast doubt upon the existence of 
irreversible processes; instead, the attempt is 
frequently made to place statistical laws in the 
foreground, and to subordinate to them laws 
hitherto regarded as dynamic, including even the 
law of gravitation. In other words, an attempt 
is made to exclude absolute law from Nature. 
And indeed, we cannot but be struck by the 
fact that the natural phenomena which we can 
investigate and measure can never be expressed 
by absolutely accurate numbers; for they in- 
evitably contain a certain inaccuracy introduced 
by the unavoidable defects of measurement itself. 
Hence it follows that we shall never succeed in 

Modern Physics 



82 

determining by measurement whether a natural 
law is absolutely valid. If we consider the ques- 
tion from the standpoint of the theory of know- 
ledge we come to the same conclusion. For if 
we cannot even prove that Nature is governed 
by law (a difficulty which we meet with at the 
very outset) a fortiori we shall be unable to 
demonstrate that such law is absolute. 

Hence from a logical point of view, we must 
admit every justification for the hypothesis that 
the only kind of law in Nature is statistical. It 
is a different question whether this assumption 
is expedient in physical research; and I feel 
strongly inclined to answer this question in the 
negative. We must consider in the first instance 
that the only type of law fully satisfying our 
desire for knowledge is the strictly dynamic type, 
while every statistical law is fundamentally 
unsatisfactory, for the simple reason that it has 
no absolute validity but admits of exceptions in 
certain cases; so that we are continually faced 
by the question what these particular exceptional 
cases are. 

Questions of this nature constitute the strongest 
argument in favour of the extension and further 
refinement of experimental methods. If it is 
assumed that statistical laws are the ultimate 



- 8 3 - 

and most profound type in existence, then there 
is no reason in theory why, when dealing with 
any particular statistical law, we should ask 
what are the causes of the variations in the 
phenomena? Actually, however, the most impor- 
tant advances in the study of atomic processes 
are due to the attempt to look for a strictly causal 
and dynamic law behind every statistical law. 

On the other hand, we may discover a law 
which has always proved absolutely valid within 
the marginal error due to measurement. In such 
a case we must admit that it will never be possible 
to prove by means of measurement that it is 
not after all of the statistical type. At the same 
time, it is of great importance whether theoretical 
considerations induce us to regard the law as 
being of the statistical, or of the dynamic, type. 
For in the first case, we should attempt to attain 
the limits of its validity by means of the con- 
tinuous refinement of our methods of measure- 
ment; in the second case, we should regard such 
attempts as useless and thus save ourselves much 
unnecessary labour. So much trouble has already 
been spent in Physics upon the solution of 
imaginary problems that such considerations are 
very far from being irrelevant. 

In my opinion, therefore, it is essential for the 



healthy development of Physics that among the 
postulates of this science we reckon, not merely 
the existence of law in general, but also the 
strictly causal character of this law. This has in 
fact almost universally been the case. Further, 
I consider it necessary to hold that the goal of 
investigation has not been reached until each 
instance of a statistical law has been analysed 
into one or more dynamic laws. I do not deny 
that the study of statistical laws is of great 
practical importance: Physics, no less than 
meteorology, geography and social science, is 
frequently compelled to make use of statistical 
laws. At the same time, however, no one will 
doubt that the alleged accidental variations of 
the climatological curves, of population statistics 
and mortality tables, are in each instance subject 
to strict causality; similarly, physicists will 
always admit that such questions are strictly 
relevant as that which asks why one of two 
neighbouring atoms of Uranium exploded many 
millions of years before the other. 

All studies dealing with the behaviour of the 
human mind are equally compelled to assume the 
existence of strict causality. The opponents of 
this view have frequently brought forward against 
it the existence of free will. In fact, however, 



-8 5 - 

there is no contradiction here; human free will 
is perfectly compatible with the universal rule 
of strict causality a view which I have had 
occasion to demonstrate in detail elsewhere. But 
as my arguments on this subject have been 
seriously misunderstood in certain quarters, and 
since this subject is surely of considerable im- 
portance, I propose to discuss it briefly here. 

The existence of strict causality implies that 
the actions, the mental processes, and especially 
the will of every individual are completely deter- 
mined at any given moment by the state of his 
mind, taken as a whole, in the previous moment, 
and by any influences acting upon him coming 
from the external world. We have no reason 
whatever for doubting the truth of this assertion. 
But the question of free will is not concerned 
with the question whether there is such a definite 
connection, but whether the person in question 
is aware of this connection. This, and this alone, 
determines whether a person can or cannot feel 
free. If a man were able to forecast his own 
future solely on the ground of causality, then 
and then only we would have to deny this 
consciousness of freedom of the will. Such a 
contingency is, however, impossible, since it con- 
tains a logical contradiction. Complete knowledge 



86 

implies that the object apprehended is not 
altered by any events taking place in the knowing 
subject; and if subject and object are identical 
this assumption does not apply. To put it more 
concretely, the knowledge of any motive or of 
any activity of will is an inner experience, from 
which a fresh motive may spring; consequently 
such an awareness increases the number of 
possible motives. But as soon as this is recognized, 
the recognition brings about a fresh act of aware- 
ness, which in its turn can generate yet another 
activity of the will. In this way the chain pro- 
ceeds, without it ever being possible to reach a 
motive which is definitely decisive for any future 
action; in other words, to reach an awareness 
which is not in its turn the occasion of a fresh 
act of will. When we look back upon a finished 
action, which we can contemplate as a whole, 
the case is completely different. Here knowledge 
no longer influences will, and hence a strictly 
causal consideration of motives and will is 
possible, at least in theory. 

If these considerations appear unintelligible 
if it is thought that a mind could completely 
grasp the causes of its present state, provided 
it were intelligent enough then such an argu- 
ment is akin to saying that a giant who is big 



-8 7 - 

enough to look down on everybody else should 
be able to look down on himself as well. The 
fact is that no person, however clever, can derive 
the decisive motives of his own conscious actions 
from the causal law alone; he requires another 
law the ethical law, for which the highest 
intelligence and the most subtle self-analysis are 
no adequate substitute. 



7 



Let us however return to Physics, from which 
these complications are excluded in advance. I 
propose now to describe the more important 
characteristics of the cunent view of the physical 
world. These characteiistics are due to the 
endeavour to find a strict causal connection, in 
the manner described above, for all physical pro- 
cesses. A cursory glance suffices to show what 
changes there have been since the beginning of 
the century; and we may say that since the days 
of Galileo and Newton, no such rapid develop- 
ment has ever been known. Incidentally we may 
point with pride to the fact that German scientists 
have played an important part in this advance. 
The occasion of this development was that 



88 

extreme refinement in measurement which is an 
essential condition of the progress of science and 
engineering; in its turn this led to the discovery 
of new facts, and hence to the revision and 
improvement of theory. Two new ideas in par- 
ticular have given modem Physics its charac- 
teristic shape. These are laid down in the Theory 
of Relativity and the Quantum hypothesis 
respectively; each in its own way is at once 
fruitful and revolutionary; but they have nothing 
in common and, in a sense, they are even 
antagonistic. 

For a time Relativity was a universal topic of 
conversation. The arguments for and against 
could be heard everywhere even in the daily 
Press, where it was championed and opposed by 
experts and by others who were very far from 
being experts. To-day things have quieted down 
a little a state of affairs which is likely to please 
nobody better than the author of the Theory 
himself; public interest appears to have become 
satisfied and to have turned to other popular 
topics. From this it might perhaps be inferred 
that the Theory of Relativity no longer plays 
any part in science. But as far as I can judge, 
the opposite is the case: for the Theory of Rela- 
tivity has now become part and parcel of the 



-8 9 - 

physical view of the world, and is taken for 
granted without any further ado. Indeed, novel 
and revolutionary as was the idea of Relativity 
(in both the Special and the General form) when 
first presented to physicists, the fact remains that 
the assertions it makes and the attacks it delivers 
were directed not against the outstanding, recog- 
nized and approved laws of Physics, but only 
against certain views which had no better sanc- 
tion than custom, deeply rooted though they 
were. These standpoints are of the kind which, 
as I have already tried to show, afford a suitable 
basis for a preliminary understanding of the facts 
of Physics; but they must be discarded as soon 
as it is found necessary to reach a more general 
and profound view of the facts. 

In this connection the idea of simultaneity is 
particularly instructive. At first glance, it seems 
to the observer that nothing could be more 
obviously true than to say that there is a definite 
meaning in asersting that two events occurring 
at two distant points (e.g. on the Earth and on 
Mars) are simultaneous. Surely every man has 
a right to traverse great distances tunelessly in 
thought, and to place two events side by side 
before the mind's eye. Now it must be empha- 
sized that the Theory of Relativity does not alter 



90 

this right in any way. If we possess sufficiently 
accurate measuring instruments, we can deter- 
mine with complete certainty whether the events 
are simultaneous; and if the time measurements 
are accurately made in different ways, and with 
different instruments which can be used to check 
each other, the same result will always be obtained. 
To this extent the Theory of Relativity has 
brought about no change whatever. 

But the Theory of Relativity does not allow 
us to assume, as a matter of course, that another 
observer who is moving relatively to ourselves 
must necessarily regard the two events as simul- 
taneous. For the thoughts and ideas of one per- 
son are not necessarily the thoughts and ideas 
of another. If the two observers proceed to discuss 
their thoughts and ideas, each will appeal to his 
own measurements; and when they do this, it 
will be found that in interpreting their respective 
measurements they started from entirely different 
assumptions. Which assumption is correct it is 
impossible to decide; and it is equally impossible 
to decide the dispute as to which of the two 
observers is in a state of rest, and which in a 
state of motion. This question, however, is of 
fundamental importance. For the rate of a clock 
alters while the clock is being moved: a fact 



9 i 

which need occasion no surprise; while from this 
it follows that the clocks of the two observers 
go at different rates. Thus we reach the conclu- 
sion that each can assert with an equal right 
that he is himself in a state of rest and that his 
time measurements are correct; and this in spite 
of the fact that the one observer regards the two 
events as simultaneous, while the other does not. 
These ideas and arguments admittedly present 
a haid task to our powers of imagination; but the 
sacrifice in clarity is negligible compared with 
the inestimable advantages which follow from 
the amazing generality and simplicity of the 
physical world- view which they render possible. 

In spite of this, some readers may still find 
themselves unable to get rid of the suspicion that 
the Theory of Relativity contains some kind of 
internal contradiction. Such readers should reflect 
that a theory, the entire content of which can be 
expressed in a single mathematical formula, can 
no more contain a contradiction than could two 
distinct conclusions following from the same 
formula. Our ideas must adjust themselves to 
the results of the formula and not conversely. 
Ultimately it is experience that must decide the 
admissibility and the importance of the Theory 
of Relativity. Indeed, the fact that experience 



92 

allows us to test its validity must be looked upon 
as the most important evidence in favour of the 
fruitfulness of the theory. Hitherto no instance 
has been recorded where the Theory conflicts 
with experience, a fact which I should like to 
emphasize in view of certain reports which have 
recently come before the public. Any one who, for 
whatever reason, considers it possible or probable 
that a conflict between the Theory and observed 
facts can be discovered, could do no better than 
co-operate in extending the Theory of Relativity 
and in pushing its conclusions as far as possible, 
since this is the only means of refuting it through 
experience. Such an undertaking is the less diffi- 
cult because the assertions made by the Theory 
of Relativity are simple and comparatively easy 
to apprehend, so that they fit into the framework 
of classical Physics without any difficulty. 

Indeed, if there were no historical objections I 
personally would not hesitate for a moment to 
include the Theory of Relativity within the body 
of classical Physics. In a manner the Theory of 
Relativity is the crowning point of Physics, since 
by merging the ideas of Time and Space it has 
also succeeded in uniting under a higher point 
of view such concepts as those of mass, energy, 
gravitation, and inertia. As the result of this 



93 

novel view we have the perfectly symmetrical 
form which the laws of the conservation of energy 
and of momentum now assume; for these laws 
follow with equal validity from the Principle of 
Least Action that most comprehensive of all 
physical laws which governs equally mechanics 
and electrodynamics. 

Now over against this strikingly imposing and 
harmonious structure there stands the Quantum 
Theory, an extraneous and threatening explosive 
body which has already succeeded in producing 
a wide and deep fissure throughout the whole of 
the structure. Unlike the Theory of Relativity, 
the Quantum Theory is not complete in itself. 
It is not a single, harmonious, and perfectly 
transparent idea, modifying the traditional facts 
and concepts of Physics by means of a change 
which, though of the utmost significance in theory, 
is practically hardly noticeable. On the contrary, 
it first arose as a means of escape from an impasse 
reached by classical Physics in one particular 
branch of its studies the explanation of the 
laws of radiant heat. It was soon seen, however, 
that it also solved with ease, or at least consider- 
ably helped to elucidate, other problems which 
were causing unmistakable difficulties to the 
classical theory, such as photoelectric phenomena, 



94 

specific heat, ionization, and chemical reactions. 
Thus it was quickly realized that the Quantum 
Theory must be regarded, not merely as a working 
hypothesis, but as a new and fundamental prin- 
ciple of Physics, whose significance becomes 
evident wherever we are dealing with rapid and 
subtle phenomena. 

Now here we are faced with a difficulty. This 
does not so much consist in the fact that the 
Quantum Theory contradicts the traditional 
views; if that were all, it follows from what has 
been said that the difficulty need not be taken 
very seriously. It arises from the fact that in 
the course of time it has become increasingly 
obvious that the Quantum Theory unequivocally 
denies certain fundamental views which are 
essential to the whole structure of the classical 
theory. Hence the introduction of the Quantum 
Theory is not a modification of the classical 
theory, as is the case with the Theory of Rela- 
tivity: it is a complete break with the classical 
theory. 

Now if the Quantum Theory were superior or 
equal to the classical theory at all points, it 
would be not only feasible but necessary to 
abandon the latter in favour of the former. This, 
however, is definitely not the case. For there are 



95 

parts of Physics, among them the wide region 
of the phenomena of interference, where the 
classical theory has proved its validity in every 
detail, even when subjected to the most delicate 
measurements; while the Quantum Theory, at 
least in its present form, is in these respects com- 
pletely useless. It is not the case that the Quantum 
Theory cannot be applied, but that, when applied, 
the results reached do not agree with experience. 

The result of this state of affairs is that at the 
present moment each theory has what may be 
called its own preserve, where it is safe from 
attack, while there is also an intermediate region 
e.g. that of the phenomena of the dispersion 
and scattering of light where the two theories 
compete with varying fortunes. The two theories 
are approximately of equal usefulness, so that 
physicists are guided in the choice of theory by 
their private predilections an uncomfortable 
and, in the long run, an intolerable state of 
affairs for anyone desirous of reaching the true 
facts. 

To illustrate this curious condition of things 
I will select a particular example from a very 
large number collected by workers in the field 
of theory and of practice. I begin by stating 
two facts. Let us imagine two fine pencils of 



_ 9 6- 

rays of violet light, produced by placing an 
opaque screen with two small holes over against 
the light which is given out from a point source. 
The two pencils of rays emerging from the holes 
can be reflected so that they meet on the surface 
of a white wall at some distance away. In this 
case the spot of light which they jointly pro- 
duce on the wall is not uniformly bright, but is 
traversed by dark lines. This is the first fact. 
The second is this if any metal that is sensitive 
to light is placed in the path of one of these rays, 
the metal will continually emit electrons with a 
velocity independent of the intensity of the light. 

Now if the intensity of the source of light is 
allowed to decrease, then in the first case, accord- 
ing to all the results hitherto obtained, the dark 
lines remain quite unchanged; it is only the 
strength of the illumination that decreases. In 
the other case, however, the velocity of the elec- 
trons emitted also remains quite unchanged, and 
the only change that takes place is that the 
emission becomes less copious. 

Now how do the theories account for these two 
facts? The first is adequately explained by the 
classical theory as follows: at every point of 
the white wall which is simultaneously illuminated 
by the two pencils of rays, the two rays which 



97 

meet at this point either strengthen or else weaken 
each other, according to the relations between 
their respective wave-lengths. The second fact 
is equally satisfactorily explained by the Quan- 
tum Theory, which maintains that the energy 
of the rays falls on the sensitive metal, not in 
a continuous flow, but in an intermittent succes- 
sion of more or less numerous, equal and indivisible 
quanta, and that each quantum, as it impinges 
on the metal, detaches one electron from the 
mass. On the other hand, all attempts have 
failed hitherto to explain the lines of interference 
by the Quantum Theory and the photoelectric 
effect by the classical theory. For if the energy 
radiated really travels only in indivisible quanta, 
then a quantum emitted from the source of light 
can pass only through one or else the other of 
the two holes in the opaque screen; while if the 
light is sufficiently feeble, it is also impossible 
for two distinct rays to impinge simultaneously 
on a single point on the white wall; hence inter- 
ference becomes impossible. In fact the lines 
invariably disappear completely, as soon as one 
of the rays is cut off. 

On the other hand, if the energy radiated from 
a point-source of light spreads out uniformly 
through space, its intensity must necessarily be 

Modem Physics 



diminished. Now it is not easy to see how the 
velocity with which an electron is emitted from 
the sensitive metal can be equally great whether 
it is subjected to very powerful or to very weak 
radiation. Naturally many attempts have been 
made to get over this difficulty. Perhaps the most 
obvious way was to assume that the energy of 
the electron emitted by the metal is not derived 
from the radiation falling on it, but that it comes 
from the interior of the metal, so that the effect 
of the radiation is merely to set it free in the 
same way as a spark sets free the latent energy 
of gunpowder. It has, however, not proved 
possible to demonstrate that there is such a 
source of energy, or even to make it appear 
plausible that there should be such a source. 
Another supposition is that, while the energy 
of the electrons is derived from the radiation 
impinging upon them, the electrons themselves 
are not actually emitted from the metal until 
this has been subjected to the illumination for 
a time sufficiently long to allow the energy 
necessary for a definite velocity to have been 
accumulated. This process, however, might take 
minutes or even hours, whereas in fact the 
phenomenon repeatedly takes place very much 
sooner. Light is thrown on the profound import- 



99 ~ 

ance of these difficulties by the fact that in highly 
influential quarters the suggestion has arisen of 
sacrificing the validity of the principle of the 
conservation of energy. This may well be described 
as a desperate remedy; in this particular instance, 
in fact, it was soon proved to be untenable by 
means of experiments. 

Hitherto, then, all attempts to understand the 
laws of the emission of electrons from the stand- 
point of the classical theory have failed. On 
the other hand these, and a number of other 
laws relating to the reciprocal action of radiation 
and matter, become immediately intelligible and 
even necessary as soon as we assume that light 
quanta travel through space in the shape of 
minute, individual structures and that, when 
impinging upon matter, they behave like really 
substantial atoms. 

We are compelled, however, to decide in favour 
of one or the other view; so that the whole 
problem obviously resolves itself into the ques- 
tion whether the radiant energy emitted from 
the source of light is divided when it leaves this 
source, so that one part of it passes through one 
of the holes in the opaque screen and the remainder 
through the other, or whether the energy passes 
in indivisible quanta alternately through each of 



100 

the two holes. Every theory of quanta must 
answer this question, and must deal with it in 
some manner or other; hitherto, however, no 
physicist has succeeded in giving a satisfactory 
answer. 

It has sometimes been suggested that the 
difficulties of the Quantum Theory do not so 
much apply to the propagation of radiation in 
free space, as to the reciprocal action which 
takes place between radiation and matter carry- 
ing an electric charge. With this opinion I cannot 
agree. The question set out above confines itself 
to the propagation of radiation, and there is no 
reference either to its causes or to its effects. 

It might indeed be asked whether we have a 
right to speak of the energy of free radiation as 
though it were something actual, since the fact 
is that all measurements invariably relate to 
events taking place in material bodies. If we 
wish to maintain the absolute validity of the 
energy principle, a standpoint which recent 
investigation renders particularly plausible, then 
there can be no doubt that we must assign to 
every field of radiation a quite definite, and more 
or less exactly calculable, amount of energy, 
which is decreased by the absorption of radiation 
and increased by its emission. The question now 



101 

is, what is the behaviour of this energy? Once 
this question is asked, it becomes plain beyond 
the possibility of doubt that we must make up 
our minds to admit certain extensions and 
generalizations of some of the primary assump- 
tions from which we are accustomed to start in 
theoretical physics, and which hitherto have 
proved their worth in every field. This becomes 
necessary in order to find a way out of the diffi- 
culty of our dilemma; and it is a result which is 
sufficiently unsatisfactory to our desire for know- 
ledge. Some consolation can be derived if we see 
that there is at any rate a possibility of solving 
the difficulty; consequently I cannot resist the 
temptation to devote a few words to discussing 
in what direction it might be possible to find a 
solution. 

The most radical method of avoiding every 
difficulty would, no doubt, consist in giving up 
the customary view which holds that radiant 
energy is localized in some manner or other; 
i.e. that at every part of a given electromagnetic 
field, a given amount of energy exists at a given 
time. If once this assumption is surrendered, the 
problem ceases to exist, simply because the 
question whether a light quantum passes through 
one or the other hole in the opaque screen ceases 



102 

to have any definite physical meaning. In my 
opinion, however, this desperate escape from the 
dilemma goes somewhat too far. For radiant 
energy as a totality possesses a definite calculable 
amount; further, the electromagnetic vector-field 
which is formed by a ray is described in all its 
optical details, and in the whole of its temporo- 
spatial behaviour, by classical electrodynamics, 
and this description agrees exactly with the facts ; 
finally the energy arises and disappears simul- 
taneously with the field. Consequently it is not 
easy to avoid the question how the distribution 
of the energy is affected by the details of the field. 
Let us decide to pursue this question as far 
as possible. Then in order to avoid the alter- 
natives with which we are faced, it might appear 
expedient to retain the fixed connection between 
the ray, or rather between the electromagnetic 
wave on the one hand, and the energy attaching 
to it on the other, but, while retaining it, to 
give it a wider and less simple meaning than it 
has in the classical theory. The latter assumes 
that every part, however small, of an electro- 
magnetic wave contains a corresponding amount 
of energy proportional to its magnitude, which 
is supposed to spread concomitantly with the 
wave. Now if for this fixed connection we substi- 



103 

tute something less rigid, it might then appear 
that the wave emitted from the source of light 
divides into any number of parts, in conformity 
with the classical theory, but that at the same 
time, in accordance with the Quantum Theory, 
the energy of the wave is concentrated at certain 
points. The necessary assumption would be that 
the energy of the wave is not intimately con- 
nected with it in its finest detail. On such an 
assumption, the phenomena of interference would 
be explained on the lines that even the weakest 
wave passes partly through one and partly 
through the other hole in the opaque screen; 
while on the other hand the photoelectric effect 
could be explained on the lines that the wave 
allows its energy to impinge on the electrons 
only in integral quanta. Here the difficulty con- 
sists in trying to imagine part of a light-wave 
without the energy appropriate to its magnitude; 
but though I admit that this is a considerable 
difficulty, I do not consider it to be essentially 
greater than that of imagining part of a body 
without the matter appropriate to its density. 
Yet we are compelled to make this latter assump- 
tion by the fact that matter loses its simple 
properties if it is subjected to continuous spatial 
sub-division, since in this case its mass ceases to 



104 

remain proportional to the space occupied by 
it, and resolves itself into a number of distinct 
molecules having a given magnitude. It might 
well be that the case is closely analogous for 
electromagnetic energy and the momentum 
attaching to it. 

Hitherto it has been the practice to look for 
the elementary laws of electromagnetic processes 
exclusively in the sphere of the infinitely small. 
Spatially and temporally all electromagnetic 
fields were divided into infinitely small parts; 
and their entire behaviour, so far as it appeared 
subject to laws, was invariably represented by 
temporo-spatial differential equations. Now in 
this respect we must radically change our views. 
For it has been discovered that these simple laws 
cease to apply after a certain stage in the process 
of subdivision has been reached, and that 
beyond this point the increasingly delicate pro- 
cesses make matters more complicated. The 
spatio-temporal magnitudes of the action become 
atomic, and we are compelled to assume the 
existence of elements or atoms of this action. It 
is indeed a sufficiently striking fact that not a 
single one of the laws where the universal quantum 
of action plays a part is expressed by means of 
a differential equation with a number of con- 



105 

tinuous variables, but that they all relate to finite 
times and finite spaces, and deal with such things 
as definite periods of oscillation, definite orbits, 
definite transitions, etc. Hence it appears that 
in order to allow for these facts we must substi- 
tute, at least in part, relations between magni- 
tudes at finite distances from each other for 
those between magnitudes infinitely close to each 
other. If this is done finite differences take the 
place of the differential, discontinuity that of 
continuity, and arithmetic that of analysis; 
though the substitution admittedly is not carried 
out radically. A radical substitution is made 
impossible if only by the claims of the wave 
theory. 

In this direction promising steps have been 
taken through the development of so-called 
Quantum Mechanics. This line of investigation 
has recently produced excellent results in the 
hands of the Gottingen school of physicists of 
Heisenberg, Born and Jordan. Later develop- 
ments will show how far we can advance towards 
a solution of the problem along the avenue 
opened by Quantum Mechanics. Even the choicest 
mathematical speculations remain in the air so 
long as they are unsubstantiated by definite 
facts of experience; and we must hope and trust 



io6 

that the experimental skill of physicists, which 
in the past has so often definitely decided ques- 
tions full of doubt and difficulty, will succeed in 
resolving the difficulties of the present obscure 
question. In any case there can be no doubt 
that the parts of the structure of classical Physics, 
which have had to be discarded as valueless under 
the pressure of the Quantum Theory, will be 
supplanted by a sounder and more adequate 
structure. 

To conclude: we have seen that the study of 
Physics, which a generation ago was one of the 
oldest and most mature of natural sciences, has 
to-day entered upon a period of storm and stress 
which promises to be the most interesting of all. 
There can be little doubt that in passing through 
this period we shall be led, not only to the dis- 
covery of new natural phenomena, but also to 
new insight into the secrets of the theory of 
knowledge. It may be that in the latter field 
many surprises await us, and that certain views, 
eclipsed at the moment, may revive and acquire 
a new significance. For this reason a careful 
study of the views and ideas of our great philo- 
sophers might prove extremely valuable in this 
direction. 

There have been times when science and philo* 



sophy were alien, if not actually antagonistic to 
each other. These times have passed. Philosophers 
have realized that they have no right to dictate 
to scientists their aims and the methods for 
attaining them; and scientists have learned that 
the starting-point of their investigations does 
not lie solely in the perceptions of the senses, 
and that science cannot exist without some small 
portion of metaphysics. Modern Physics impresses 
us particularly with the truth of the old doctrine 
which teaches that there are realities existing 
apart from our sense-perceptions, and that there 
are problems and conflicts where these realities 
are of greater value for us than the richest 
treasures of the world of experience. 



INDEX 



ACTION, 20, 30, 41, 93, 104 
Axiomatic school of physi- 
cists, II, 22 

Bohr, Niels, 31 
Born, 43, 105 
Brownian movement, 80 

Causality and free will, 

84 sqq., 87 

Configuration point, 38, 41 
Configuration space, 2 7 sqq . , 

33 $4 36, 44 
Conservation, 22, 99 
Copernicus, 64 

de Broglie, 43 
Determinism, 48, 54 
Dirac, 43 

Einstein, 18 

Electricity, 16 

Electron, idsq., 21, 31 sq., 

4<>, 42, 57 
Energetics, theory of, 76, 

81 

Energy, 19, 22, 26, 92 
Ether, 72 
Evolution, 10, 13, 15 

Force, Newton's concept 

of, 66 sqq., 69 sq. 
Franck, 56 



Freedom, 84 sq. 
Fresnel, 46 

Galileo, 66, 87 
Gauss, 1 8 
Gravitation, 19, 78, 92 

Hamilton, 28, 41 
Heat, 72, 75 sqq., 80, 93 
Heisenberg, 38 sq., 43, 47, 

49, 105 
Hertz, 57 
Hydrogen, i6sq. 
Hypothesis, 60 sq. 

Indeterminism, 47, 50, 52 
Irreversible processes, 77, 
84 

Jacobi, 41 
Jordan, 43, 105 

Kepler, 64 sq. 
Kurlbaum, 56 



Lagrange, 28 
Law, ii sq. t 

8 1 sq. 

Leibniz, 52 
Lenard, 56 
Light, 95 sq. 
Lorentz, 46 
Lummer, 56 



74, 



110 



Magnitude, 12, 45 sq., 77 
Mass, 19, 92 

Mathematics, 14, 18, 43, 74 
Maxwell, 42 
Measurement, 7, 60, 73, 

77 ^. 

Michelson, 56 
Mind, 84 
Momentum, 19 

Nature, 8, 59, Si sq. 
Newton, 28, 64 sqq., 67, 
70 sq., 87 

Osmotic pressure, 70 

Particle, 26 
Perception, 7 
Photoelectric effect, 97, 

103 

Positivism, n sq. 
Potential, 68 
Pringsheim, 56 
Probability, 47 
Protons, 1 6 sq., 21, 42 
Ptolemy, 64 

Quantum Mechanics, 45 
Quantum of Action, 20 sqq., 

23, 30, 38, 41, 104 
Quantum Theory, 17, 20 
sqq., 26, 29, 37 sq., 88, 
93 sqq., 97 $M-> IO 3 Io6 



Reality, 8, 15 

Relativity theory, 17 sqq., 
20, 22, 27, 42, 46, 
55 sq., 88 sqq., gi sqq., 

94 

Reversible processes, 77-84 
Riemann, 19 
Ruben, 56 

Schrodinger, 28, 41, 43 
Sense-data, 7 
Sense-perception, 7 
Sense, the world of, 14 sq. 
Simultaneity, 89 sq. 
Space, 18, 25, 55, 63, 92 
Statistics, 81 sq. 
Substance, 71 sq. 

Thermodynamics, 22 
Time, 18, 55, 63, 92 

Uncertainty principle, 38, 

47> 49, 52 
Uranium, 17 sq. 

van't Hof , 70 
Vector, 27 

Wave groups, 37 
Will, 84 sq. 

Work, mechanical concept 
of, 67 sq. 




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